Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 23.2s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. fma-define N/A

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

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

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

4. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

   4. cos-atan N/A

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

   5. un-div-inv N/A

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

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

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

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

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

   8. hypot-1-def N/A

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

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

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

   10. associate-/l/ N/A

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

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

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

   12. *-commutative N/A

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

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

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

   14. tan-lowering-tan.f64 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 98.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. fma-define N/A

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

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

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

4. Applied egg-rr 99.8%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. sin-atan N/A

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

   4. metadata-eval N/A

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

   5. associate-/r* N/A

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

   6. div-inv N/A

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

   7. associate-*r/ N/A

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

   8. div-inv N/A

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

   9. associate-*l* N/A

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

6. Applied egg-rr 77.5%

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

   1. associate-*l/ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

   5. clear-num N/A

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

   6. associate-*l/ N/A

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

8. Applied egg-rr 99.8%

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

9. Taylor expanded in ew around inf

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\color{blue}{ew}, \mathsf{sin.f64}\left(t\right), \mathsf{/.f64}\left(eh, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(t\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]
10. Step-by-step derivation

11. Simplified 97.4%

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

Alternative 4: 86.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ t_2 := \frac{eh}{ew \cdot \tan t}\\ t_3 := \mathsf{hypot}\left(1, t\_2\right)\\ t_4 := \left|\frac{ew \cdot \sin t + t\_2 \cdot t\_1}{t\_3}\right|\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{-15}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-50}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot t\right) \cdot \cos \tan^{-1} t\_2\right|\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+146}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{ew}{t\_3}, \sin t, eh \cdot \left(t\_1 \cdot \frac{\frac{ew}{eh}}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t)))
        (t_2 (/ eh (* ew (tan t))))
        (t_3 (hypot 1.0 t_2))
        (t_4 (fabs (/ (+ (* ew (sin t)) (* t_2 t_1)) t_3))))
   (if (<= t -2.7e-15)
     t_4
     (if (<= t 1.35e-50)
       (fabs
        (+ (* eh (sin (atan (/ eh (* ew t))))) (* (* ew t) (cos (atan t_2)))))
       (if (<= t 1.15e+146)
         (fabs (fma (/ ew t_3) (sin t) (* eh (* t_1 (/ (/ ew eh) ew)))))
         t_4)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double t_2 = eh / (ew * tan(t));
	double t_3 = hypot(1.0, t_2);
	double t_4 = fabs((((ew * sin(t)) + (t_2 * t_1)) / t_3));
	double tmp;
	if (t <= -2.7e-15) {
		tmp = t_4;
	} else if (t <= 1.35e-50) {
		tmp = fabs(((eh * sin(atan((eh / (ew * t))))) + ((ew * t) * cos(atan(t_2)))));
	} else if (t <= 1.15e+146) {
		tmp = fabs(fma((ew / t_3), sin(t), (eh * (t_1 * ((ew / eh) / ew)))));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	t_2 = Float64(eh / Float64(ew * tan(t)))
	t_3 = hypot(1.0, t_2)
	t_4 = abs(Float64(Float64(Float64(ew * sin(t)) + Float64(t_2 * t_1)) / t_3))
	tmp = 0.0
	if (t <= -2.7e-15)
		tmp = t_4;
	elseif (t <= 1.35e-50)
		tmp = abs(Float64(Float64(eh * sin(atan(Float64(eh / Float64(ew * t))))) + Float64(Float64(ew * t) * cos(atan(t_2)))));
	elseif (t <= 1.15e+146)
		tmp = abs(fma(Float64(ew / t_3), sin(t), Float64(eh * Float64(t_1 * Float64(Float64(ew / eh) / ew)))));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[1.0 ^ 2 + t$95$2 ^ 2], $MachinePrecision]}, Block[{t$95$4 = N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -2.7e-15], t$95$4, If[LessEqual[t, 1.35e-50], N[Abs[N[(N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * t), $MachinePrecision] * N[Cos[N[ArcTan[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.15e+146], N[Abs[N[(N[(ew / t$95$3), $MachinePrecision] * N[Sin[t], $MachinePrecision] + N[(eh * N[(t$95$1 * N[(N[(ew / eh), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.70000000000000009e-15 or 1.15e146 < t

   1. Initial program 99.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. cos-atan N/A

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

      5. un-div-inv N/A

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

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

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

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

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

      8. hypot-1-def N/A

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

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

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

      10. associate-/l/ N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. tan-lowering-tan.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   6. Applied egg-rr 85.9%

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

      3. div-fabs N/A

         \[\leadsto \left|\frac{\frac{eh \cdot \cos t}{\frac{ew}{\frac{eh}{\tan t}}} + \sin t \cdot ew}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right| \]

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

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

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

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

   8. Applied egg-rr 86.3%

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

   if -2.70000000000000009e-15 < t < 1.35e-50

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. tan-lowering-tan.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\color{blue}{\left(\frac{eh}{ew \cdot t}\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right), \mathsf{*.f64}\left(t, ew\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   if 1.35e-50 < t < 1.15e146

   1. Initial program 99.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. fma-define N/A

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

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

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

   4. Applied egg-rr 99.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. sin-atan N/A

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

      4. metadata-eval N/A

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

      5. associate-/r* N/A

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

      6. div-inv N/A

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

      7. associate-*r/ N/A

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

      8. div-inv N/A

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

      9. associate-*l* N/A

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

   6. Applied egg-rr 94.8%

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

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(ew, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(t\right), \mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{ew}{eh}\right)}, ew\right), \mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 88.3%

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

   9. Simplified 88.3%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-15}:\\ \;\;\;\;\left|\frac{ew \cdot \sin t + \frac{eh}{ew \cdot \tan t} \cdot \left(eh \cdot \cos t\right)}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}\right|\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-50}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+146}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{ew}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}, \sin t, eh \cdot \left(\left(eh \cdot \cos t\right) \cdot \frac{\frac{ew}{eh}}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{ew \cdot \sin t + \frac{eh}{ew \cdot \tan t} \cdot \left(eh \cdot \cos t\right)}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew \cdot \tan t}\\ t_2 := \left|\frac{ew \cdot \sin t + t\_1 \cdot \left(eh \cdot \cos t\right)}{\mathsf{hypot}\left(1, t\_1\right)}\right|\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{-15}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot t\right) \cdot \cos \tan^{-1} t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew (tan t))))
        (t_2
         (fabs (/ (+ (* ew (sin t)) (* t_1 (* eh (cos t)))) (hypot 1.0 t_1)))))
   (if (<= t -2.7e-15)
     t_2
     (if (<= t 1.6e-6)
       (fabs
        (+ (* eh (sin (atan (/ eh (* ew t))))) (* (* ew t) (cos (atan t_1)))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * tan(t));
	double t_2 = fabs((((ew * sin(t)) + (t_1 * (eh * cos(t)))) / hypot(1.0, t_1)));
	double tmp;
	if (t <= -2.7e-15) {
		tmp = t_2;
	} else if (t <= 1.6e-6) {
		tmp = fabs(((eh * sin(atan((eh / (ew * t))))) + ((ew * t) * cos(atan(t_1)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * Math.tan(t));
	double t_2 = Math.abs((((ew * Math.sin(t)) + (t_1 * (eh * Math.cos(t)))) / Math.hypot(1.0, t_1)));
	double tmp;
	if (t <= -2.7e-15) {
		tmp = t_2;
	} else if (t <= 1.6e-6) {
		tmp = Math.abs(((eh * Math.sin(Math.atan((eh / (ew * t))))) + ((ew * t) * Math.cos(Math.atan(t_1)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh / (ew * math.tan(t))
	t_2 = math.fabs((((ew * math.sin(t)) + (t_1 * (eh * math.cos(t)))) / math.hypot(1.0, t_1)))
	tmp = 0
	if t <= -2.7e-15:
		tmp = t_2
	elif t <= 1.6e-6:
		tmp = math.fabs(((eh * math.sin(math.atan((eh / (ew * t))))) + ((ew * t) * math.cos(math.atan(t_1)))))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * tan(t)))
	t_2 = abs(Float64(Float64(Float64(ew * sin(t)) + Float64(t_1 * Float64(eh * cos(t)))) / hypot(1.0, t_1)))
	tmp = 0.0
	if (t <= -2.7e-15)
		tmp = t_2;
	elseif (t <= 1.6e-6)
		tmp = abs(Float64(Float64(eh * sin(atan(Float64(eh / Float64(ew * t))))) + Float64(Float64(ew * t) * cos(atan(t_1)))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh / (ew * tan(t));
	t_2 = abs((((ew * sin(t)) + (t_1 * (eh * cos(t)))) / hypot(1.0, t_1)));
	tmp = 0.0;
	if (t <= -2.7e-15)
		tmp = t_2;
	elseif (t <= 1.6e-6)
		tmp = abs(((eh * sin(atan((eh / (ew * t))))) + ((ew * t) * cos(atan(t_1)))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -2.7e-15], t$95$2, If[LessEqual[t, 1.6e-6], N[Abs[N[(N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * t), $MachinePrecision] * N[Cos[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.70000000000000009e-15 or 1.5999999999999999e-6 < t

   1. Initial program 99.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. cos-atan N/A

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

      5. un-div-inv N/A

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

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

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

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

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

      8. hypot-1-def N/A

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

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

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

      10. associate-/l/ N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. tan-lowering-tan.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   6. Applied egg-rr 82.5%

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

      1. fabs-div N/A

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

      2. clear-num N/A

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

      3. div-fabs N/A

         \[\leadsto \left|\frac{\frac{eh \cdot \cos t}{\frac{ew}{\frac{eh}{\tan t}}} + \sin t \cdot ew}{\sqrt{1 \cdot 1 + \frac{eh}{ew \cdot \tan t} \cdot \frac{eh}{ew \cdot \tan t}}}\right| \]

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

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

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

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

   8. Applied egg-rr 82.8%

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

   if -2.70000000000000009e-15 < t < 1.5999999999999999e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. tan-lowering-tan.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\color{blue}{\left(\frac{eh}{ew \cdot t}\right)}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right), \mathsf{*.f64}\left(t, ew\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-15}:\\ \;\;\;\;\left|\frac{ew \cdot \sin t + \frac{eh}{ew \cdot \tan t} \cdot \left(eh \cdot \cos t\right)}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}\right|\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{ew \cdot \sin t + \frac{eh}{ew \cdot \tan t} \cdot \left(eh \cdot \cos t\right)}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew \cdot \tan t}\\ t_2 := \left|\frac{ew \cdot \sin t + eh \cdot t\_1}{\mathsf{hypot}\left(1, t\_1\right)}\right|\\ \mathbf{if}\;ew \leq -6.5 \cdot 10^{+35}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 10^{+136}:\\ \;\;\;\;\left|eh \cdot \cos t + \left(ew \cdot \frac{0.5}{eh}\right) \cdot \left(ew \cdot \left(\tan t \cdot \sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew (tan t))))
        (t_2 (fabs (/ (+ (* ew (sin t)) (* eh t_1)) (hypot 1.0 t_1)))))
   (if (<= ew -6.5e+35)
     t_2
     (if (<= ew 1e+136)
       (fabs
        (+ (* eh (cos t)) (* (* ew (/ 0.5 eh)) (* ew (* (tan t) (sin t))))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * tan(t));
	double t_2 = fabs((((ew * sin(t)) + (eh * t_1)) / hypot(1.0, t_1)));
	double tmp;
	if (ew <= -6.5e+35) {
		tmp = t_2;
	} else if (ew <= 1e+136) {
		tmp = fabs(((eh * cos(t)) + ((ew * (0.5 / eh)) * (ew * (tan(t) * sin(t))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * Math.tan(t));
	double t_2 = Math.abs((((ew * Math.sin(t)) + (eh * t_1)) / Math.hypot(1.0, t_1)));
	double tmp;
	if (ew <= -6.5e+35) {
		tmp = t_2;
	} else if (ew <= 1e+136) {
		tmp = Math.abs(((eh * Math.cos(t)) + ((ew * (0.5 / eh)) * (ew * (Math.tan(t) * Math.sin(t))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh / (ew * math.tan(t))
	t_2 = math.fabs((((ew * math.sin(t)) + (eh * t_1)) / math.hypot(1.0, t_1)))
	tmp = 0
	if ew <= -6.5e+35:
		tmp = t_2
	elif ew <= 1e+136:
		tmp = math.fabs(((eh * math.cos(t)) + ((ew * (0.5 / eh)) * (ew * (math.tan(t) * math.sin(t))))))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * tan(t)))
	t_2 = abs(Float64(Float64(Float64(ew * sin(t)) + Float64(eh * t_1)) / hypot(1.0, t_1)))
	tmp = 0.0
	if (ew <= -6.5e+35)
		tmp = t_2;
	elseif (ew <= 1e+136)
		tmp = abs(Float64(Float64(eh * cos(t)) + Float64(Float64(ew * Float64(0.5 / eh)) * Float64(ew * Float64(tan(t) * sin(t))))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh / (ew * tan(t));
	t_2 = abs((((ew * sin(t)) + (eh * t_1)) / hypot(1.0, t_1)));
	tmp = 0.0;
	if (ew <= -6.5e+35)
		tmp = t_2;
	elseif (ew <= 1e+136)
		tmp = abs(((eh * cos(t)) + ((ew * (0.5 / eh)) * (ew * (tan(t) * sin(t))))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(eh * t$95$1), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -6.5e+35], t$95$2, If[LessEqual[ew, 1e+136], N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[(0.5 / eh), $MachinePrecision]), $MachinePrecision] * N[(ew * N[(N[Tan[t], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -6.5000000000000003e35 or 1.00000000000000006e136 < ew

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. fma-define N/A

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

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

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

   4. Applied egg-rr 99.8%

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

   6. Applied egg-rr 85.5%

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

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{eh}, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(t\right), \mathsf{/.f64}\left(eh, ew\right)\right)\right), \mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 79.2%

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

       1. fabs-div N/A

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

       2. metadata-eval N/A

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

       3. remove-double-div N/A

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

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

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

       5. *-commutative N/A

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

       6. un-div-inv N/A

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

   11. Applied egg-rr 79.5%

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

   if -6.5000000000000003e35 < ew < 1.00000000000000006e136

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. sin-atan N/A

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

      3. div-inv N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. fma-define N/A

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

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

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

   4. Applied egg-rr 53.5%

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

   5. Taylor expanded in ew around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

      15. cos-lowering-cos.f64 80.6%

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

   7. Simplified 80.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. tan-quot N/A

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

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

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

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

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

      15. tan-lowering-tan.f64 80.8%

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

   9. Applied egg-rr 80.8%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -6.5 \cdot 10^{+35}:\\ \;\;\;\;\left|\frac{ew \cdot \sin t + eh \cdot \frac{eh}{ew \cdot \tan t}}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}\right|\\ \mathbf{elif}\;ew \leq 10^{+136}:\\ \;\;\;\;\left|eh \cdot \cos t + \left(ew \cdot \frac{0.5}{eh}\right) \cdot \left(ew \cdot \left(\tan t \cdot \sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{ew \cdot \sin t + eh \cdot \frac{eh}{ew \cdot \tan t}}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\left|\frac{1}{\frac{1}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot t}\right)} \cdot \left(ew \cdot \sin t + \frac{eh}{\frac{\tan t}{\frac{eh}{ew}}}\right)}\right|}\\ \mathbf{if}\;ew \leq -5.6 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 10^{+136}:\\ \;\;\;\;\left|eh \cdot \cos t + \left(ew \cdot \frac{0.5}{eh}\right) \cdot \left(ew \cdot \left(\tan t \cdot \sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (/
          1.0
          (fabs
           (/
            1.0
            (*
             (/ 1.0 (hypot 1.0 (/ eh (* ew t))))
             (+ (* ew (sin t)) (/ eh (/ (tan t) (/ eh ew))))))))))
   (if (<= ew -5.6e+37)
     t_1
     (if (<= ew 1e+136)
       (fabs
        (+ (* eh (cos t)) (* (* ew (/ 0.5 eh)) (* ew (* (tan t) (sin t))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = 1.0 / fabs((1.0 / ((1.0 / hypot(1.0, (eh / (ew * t)))) * ((ew * sin(t)) + (eh / (tan(t) / (eh / ew)))))));
	double tmp;
	if (ew <= -5.6e+37) {
		tmp = t_1;
	} else if (ew <= 1e+136) {
		tmp = fabs(((eh * cos(t)) + ((ew * (0.5 / eh)) * (ew * (tan(t) * sin(t))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = 1.0 / Math.abs((1.0 / ((1.0 / Math.hypot(1.0, (eh / (ew * t)))) * ((ew * Math.sin(t)) + (eh / (Math.tan(t) / (eh / ew)))))));
	double tmp;
	if (ew <= -5.6e+37) {
		tmp = t_1;
	} else if (ew <= 1e+136) {
		tmp = Math.abs(((eh * Math.cos(t)) + ((ew * (0.5 / eh)) * (ew * (Math.tan(t) * Math.sin(t))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = 1.0 / math.fabs((1.0 / ((1.0 / math.hypot(1.0, (eh / (ew * t)))) * ((ew * math.sin(t)) + (eh / (math.tan(t) / (eh / ew)))))))
	tmp = 0
	if ew <= -5.6e+37:
		tmp = t_1
	elif ew <= 1e+136:
		tmp = math.fabs(((eh * math.cos(t)) + ((ew * (0.5 / eh)) * (ew * (math.tan(t) * math.sin(t))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(1.0 / abs(Float64(1.0 / Float64(Float64(1.0 / hypot(1.0, Float64(eh / Float64(ew * t)))) * Float64(Float64(ew * sin(t)) + Float64(eh / Float64(tan(t) / Float64(eh / ew))))))))
	tmp = 0.0
	if (ew <= -5.6e+37)
		tmp = t_1;
	elseif (ew <= 1e+136)
		tmp = abs(Float64(Float64(eh * cos(t)) + Float64(Float64(ew * Float64(0.5 / eh)) * Float64(ew * Float64(tan(t) * sin(t))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = 1.0 / abs((1.0 / ((1.0 / hypot(1.0, (eh / (ew * t)))) * ((ew * sin(t)) + (eh / (tan(t) / (eh / ew)))))));
	tmp = 0.0;
	if (ew <= -5.6e+37)
		tmp = t_1;
	elseif (ew <= 1e+136)
		tmp = abs(((eh * cos(t)) + ((ew * (0.5 / eh)) * (ew * (tan(t) * sin(t))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(1.0 / N[Abs[N[(1.0 / N[(N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(eh / N[(N[Tan[t], $MachinePrecision] / N[(eh / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -5.6e+37], t$95$1, If[LessEqual[ew, 1e+136], N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[(0.5 / eh), $MachinePrecision]), $MachinePrecision] * N[(ew * N[(N[Tan[t], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.5999999999999996e37 or 1.00000000000000006e136 < ew

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. fma-define N/A

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

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

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

   4. Applied egg-rr 99.8%

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

   6. Applied egg-rr 85.5%

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

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{eh}, \mathsf{/.f64}\left(\mathsf{tan.f64}\left(t\right), \mathsf{/.f64}\left(eh, ew\right)\right)\right), \mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 79.2%

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

   10. Taylor expanded in t around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 78.6%

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

   12. Simplified 78.6%

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

   if -5.5999999999999996e37 < ew < 1.00000000000000006e136

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. sin-atan N/A

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

      3. div-inv N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. fma-define N/A

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

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

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

   4. Applied egg-rr 53.5%

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

   5. Taylor expanded in ew around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

      15. cos-lowering-cos.f64 80.6%

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

   7. Simplified 80.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. tan-quot N/A

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

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

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

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

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

      15. tan-lowering-tan.f64 80.8%

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

   9. Applied egg-rr 80.8%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5.6 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\frac{1}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot t}\right)} \cdot \left(ew \cdot \sin t + \frac{eh}{\frac{\tan t}{\frac{eh}{ew}}}\right)}\right|}\\ \mathbf{elif}\;ew \leq 10^{+136}:\\ \;\;\;\;\left|eh \cdot \cos t + \left(ew \cdot \frac{0.5}{eh}\right) \cdot \left(ew \cdot \left(\tan t \cdot \sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\frac{1}{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot t}\right)} \cdot \left(ew \cdot \sin t + \frac{eh}{\frac{\tan t}{\frac{eh}{ew}}}\right)}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ \mathbf{if}\;ew \leq -2.6 \cdot 10^{+196}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{elif}\;ew \leq 1.1 \cdot 10^{+136}:\\ \;\;\;\;\left|eh \cdot \cos t + \left(ew \cdot \frac{0.5}{eh}\right) \cdot \left(ew \cdot \left(\tan t \cdot \sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}{t\_1 + \frac{\frac{eh \cdot eh}{ew}}{t}}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))))
   (if (<= ew -2.6e+196)
     (fabs t_1)
     (if (<= ew 1.1e+136)
       (fabs
        (+ (* eh (cos t)) (* (* ew (/ 0.5 eh)) (* ew (* (tan t) (sin t))))))
       (/
        1.0
        (fabs
         (/
          (hypot 1.0 (/ eh (* ew (tan t))))
          (+ t_1 (/ (/ (* eh eh) ew) t)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double tmp;
	if (ew <= -2.6e+196) {
		tmp = fabs(t_1);
	} else if (ew <= 1.1e+136) {
		tmp = fabs(((eh * cos(t)) + ((ew * (0.5 / eh)) * (ew * (tan(t) * sin(t))))));
	} else {
		tmp = 1.0 / fabs((hypot(1.0, (eh / (ew * tan(t)))) / (t_1 + (((eh * eh) / ew) / t))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.sin(t);
	double tmp;
	if (ew <= -2.6e+196) {
		tmp = Math.abs(t_1);
	} else if (ew <= 1.1e+136) {
		tmp = Math.abs(((eh * Math.cos(t)) + ((ew * (0.5 / eh)) * (ew * (Math.tan(t) * Math.sin(t))))));
	} else {
		tmp = 1.0 / Math.abs((Math.hypot(1.0, (eh / (ew * Math.tan(t)))) / (t_1 + (((eh * eh) / ew) / t))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.sin(t)
	tmp = 0
	if ew <= -2.6e+196:
		tmp = math.fabs(t_1)
	elif ew <= 1.1e+136:
		tmp = math.fabs(((eh * math.cos(t)) + ((ew * (0.5 / eh)) * (ew * (math.tan(t) * math.sin(t))))))
	else:
		tmp = 1.0 / math.fabs((math.hypot(1.0, (eh / (ew * math.tan(t)))) / (t_1 + (((eh * eh) / ew) / t))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	tmp = 0.0
	if (ew <= -2.6e+196)
		tmp = abs(t_1);
	elseif (ew <= 1.1e+136)
		tmp = abs(Float64(Float64(eh * cos(t)) + Float64(Float64(ew * Float64(0.5 / eh)) * Float64(ew * Float64(tan(t) * sin(t))))));
	else
		tmp = Float64(1.0 / abs(Float64(hypot(1.0, Float64(eh / Float64(ew * tan(t)))) / Float64(t_1 + Float64(Float64(Float64(eh * eh) / ew) / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * sin(t);
	tmp = 0.0;
	if (ew <= -2.6e+196)
		tmp = abs(t_1);
	elseif (ew <= 1.1e+136)
		tmp = abs(((eh * cos(t)) + ((ew * (0.5 / eh)) * (ew * (tan(t) * sin(t))))));
	else
		tmp = 1.0 / abs((hypot(1.0, (eh / (ew * tan(t)))) / (t_1 + (((eh * eh) / ew) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -2.6e+196], N[Abs[t$95$1], $MachinePrecision], If[LessEqual[ew, 1.1e+136], N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[(0.5 / eh), $MachinePrecision]), $MachinePrecision] * N[(ew * N[(N[Tan[t], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Abs[N[(N[Sqrt[1.0 ^ 2 + N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[(t$95$1 + N[(N[(N[(eh * eh), $MachinePrecision] / ew), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -2.60000000000000012e196

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. fma-define N/A

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

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

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in ew around inf

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

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

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

      2. sin-lowering-sin.f64 82.6%

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

   7. Simplified 82.6%

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

   if -2.60000000000000012e196 < ew < 1.1e136

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. sin-atan N/A

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

      3. div-inv N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. fma-define N/A

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

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

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in ew around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

      15. cos-lowering-cos.f64 75.0%

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

   7. Simplified 75.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. tan-quot N/A

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

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

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

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

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

      15. tan-lowering-tan.f64 77.8%

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

   9. Applied egg-rr 77.8%

      \[\leadsto \left|eh \cdot \cos t + \color{blue}{\left(ew \cdot \frac{0.5}{eh}\right) \cdot \left(\left(\sin t \cdot \tan t\right) \cdot ew\right)}\right| \]

   if 1.1e136 < ew

   1. Initial program 99.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. cos-atan N/A

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

      5. un-div-inv N/A

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

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

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

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

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

      8. hypot-1-def N/A

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

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

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

      10. associate-/l/ N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. tan-lowering-tan.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   6. Applied egg-rr 86.0%

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

   7. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{hypot.f64}\left(1, \mathsf{/.f64}\left(eh, \mathsf{*.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\left(\frac{{eh}^{2}}{ew \cdot t}\right)}, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), ew\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. associate-/r* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 76.5%

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

   9. Simplified 76.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.6 \cdot 10^{+196}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{elif}\;ew \leq 1.1 \cdot 10^{+136}:\\ \;\;\;\;\left|eh \cdot \cos t + \left(ew \cdot \frac{0.5}{eh}\right) \cdot \left(ew \cdot \left(\tan t \cdot \sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{\mathsf{hypot}\left(1, \frac{eh}{ew \cdot \tan t}\right)}{ew \cdot \sin t + \frac{\frac{eh \cdot eh}{ew}}{t}}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ t_2 := ew \cdot \sin t\\ \mathbf{if}\;ew \leq -6.2 \cdot 10^{+190}:\\ \;\;\;\;\left|t\_2\right|\\ \mathbf{elif}\;ew \leq 1.08 \cdot 10^{+136}:\\ \;\;\;\;\left|t\_1 + \left(ew \cdot \frac{0.5}{eh}\right) \cdot \left(ew \cdot \left(\tan t \cdot \sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{t\_2 + \frac{t\_1}{\frac{\tan t}{\frac{eh}{ew}}}}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))) (t_2 (* ew (sin t))))
   (if (<= ew -6.2e+190)
     (fabs t_2)
     (if (<= ew 1.08e+136)
       (fabs (+ t_1 (* (* ew (/ 0.5 eh)) (* ew (* (tan t) (sin t))))))
       (/ 1.0 (fabs (/ 1.0 (+ t_2 (/ t_1 (/ (tan t) (/ eh ew)))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double t_2 = ew * sin(t);
	double tmp;
	if (ew <= -6.2e+190) {
		tmp = fabs(t_2);
	} else if (ew <= 1.08e+136) {
		tmp = fabs((t_1 + ((ew * (0.5 / eh)) * (ew * (tan(t) * sin(t))))));
	} else {
		tmp = 1.0 / fabs((1.0 / (t_2 + (t_1 / (tan(t) / (eh / ew))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = eh * cos(t)
    t_2 = ew * sin(t)
    if (ew <= (-6.2d+190)) then
        tmp = abs(t_2)
    else if (ew <= 1.08d+136) then
        tmp = abs((t_1 + ((ew * (0.5d0 / eh)) * (ew * (tan(t) * sin(t))))))
    else
        tmp = 1.0d0 / abs((1.0d0 / (t_2 + (t_1 / (tan(t) / (eh / ew))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.cos(t);
	double t_2 = ew * Math.sin(t);
	double tmp;
	if (ew <= -6.2e+190) {
		tmp = Math.abs(t_2);
	} else if (ew <= 1.08e+136) {
		tmp = Math.abs((t_1 + ((ew * (0.5 / eh)) * (ew * (Math.tan(t) * Math.sin(t))))));
	} else {
		tmp = 1.0 / Math.abs((1.0 / (t_2 + (t_1 / (Math.tan(t) / (eh / ew))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.cos(t)
	t_2 = ew * math.sin(t)
	tmp = 0
	if ew <= -6.2e+190:
		tmp = math.fabs(t_2)
	elif ew <= 1.08e+136:
		tmp = math.fabs((t_1 + ((ew * (0.5 / eh)) * (ew * (math.tan(t) * math.sin(t))))))
	else:
		tmp = 1.0 / math.fabs((1.0 / (t_2 + (t_1 / (math.tan(t) / (eh / ew))))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	t_2 = Float64(ew * sin(t))
	tmp = 0.0
	if (ew <= -6.2e+190)
		tmp = abs(t_2);
	elseif (ew <= 1.08e+136)
		tmp = abs(Float64(t_1 + Float64(Float64(ew * Float64(0.5 / eh)) * Float64(ew * Float64(tan(t) * sin(t))))));
	else
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(t_2 + Float64(t_1 / Float64(tan(t) / Float64(eh / ew)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * cos(t);
	t_2 = ew * sin(t);
	tmp = 0.0;
	if (ew <= -6.2e+190)
		tmp = abs(t_2);
	elseif (ew <= 1.08e+136)
		tmp = abs((t_1 + ((ew * (0.5 / eh)) * (ew * (tan(t) * sin(t))))));
	else
		tmp = 1.0 / abs((1.0 / (t_2 + (t_1 / (tan(t) / (eh / ew))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -6.2e+190], N[Abs[t$95$2], $MachinePrecision], If[LessEqual[ew, 1.08e+136], N[Abs[N[(t$95$1 + N[(N[(ew * N[(0.5 / eh), $MachinePrecision]), $MachinePrecision] * N[(ew * N[(N[Tan[t], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Abs[N[(1.0 / N[(t$95$2 + N[(t$95$1 / N[(N[Tan[t], $MachinePrecision] / N[(eh / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -6.2000000000000003e190

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. fma-define N/A

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

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

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in ew around inf

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

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

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

      2. sin-lowering-sin.f64 82.6%

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

   7. Simplified 82.6%

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

   if -6.2000000000000003e190 < ew < 1.07999999999999994e136

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. sin-atan N/A

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

      3. div-inv N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. fma-define N/A

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

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

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in ew around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

      15. cos-lowering-cos.f64 75.0%

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

   7. Simplified 75.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. tan-quot N/A

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

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

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

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

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

      15. tan-lowering-tan.f64 77.8%

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

   9. Applied egg-rr 77.8%

      \[\leadsto \left|eh \cdot \cos t + \color{blue}{\left(ew \cdot \frac{0.5}{eh}\right) \cdot \left(\left(\sin t \cdot \tan t\right) \cdot ew\right)}\right| \]

   if 1.07999999999999994e136 < ew

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. fma-define N/A

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

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

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

   4. Applied egg-rr 99.7%

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

   6. Applied egg-rr 90.6%

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

   7. Taylor expanded in eh around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(\mathsf{tan.f64}\left(t\right), \mathsf{/.f64}\left(eh, ew\right)\right)\right), \mathsf{*.f64}\left(ew, \mathsf{sin.f64}\left(t\right)\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 74.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -6.2 \cdot 10^{+190}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{elif}\;ew \leq 1.08 \cdot 10^{+136}:\\ \;\;\;\;\left|eh \cdot \cos t + \left(ew \cdot \frac{0.5}{eh}\right) \cdot \left(ew \cdot \left(\tan t \cdot \sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew \cdot \sin t + \frac{eh \cdot \cos t}{\frac{\tan t}{\frac{eh}{ew}}}}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ t_2 := ew \cdot \sin t\\ \mathbf{if}\;ew \leq -1.25 \cdot 10^{+191}:\\ \;\;\;\;\left|t\_2\right|\\ \mathbf{elif}\;ew \leq 1.35 \cdot 10^{+136}:\\ \;\;\;\;\left|t\_1 + \left(ew \cdot \frac{0.5}{eh}\right) \cdot \left(ew \cdot \left(\tan t \cdot \sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{t\_2 + \frac{t\_1}{\frac{ew}{\frac{eh}{\tan t}}}}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))) (t_2 (* ew (sin t))))
   (if (<= ew -1.25e+191)
     (fabs t_2)
     (if (<= ew 1.35e+136)
       (fabs (+ t_1 (* (* ew (/ 0.5 eh)) (* ew (* (tan t) (sin t))))))
       (/ 1.0 (fabs (/ 1.0 (+ t_2 (/ t_1 (/ ew (/ eh (tan t))))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double t_2 = ew * sin(t);
	double tmp;
	if (ew <= -1.25e+191) {
		tmp = fabs(t_2);
	} else if (ew <= 1.35e+136) {
		tmp = fabs((t_1 + ((ew * (0.5 / eh)) * (ew * (tan(t) * sin(t))))));
	} else {
		tmp = 1.0 / fabs((1.0 / (t_2 + (t_1 / (ew / (eh / tan(t)))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = eh * cos(t)
    t_2 = ew * sin(t)
    if (ew <= (-1.25d+191)) then
        tmp = abs(t_2)
    else if (ew <= 1.35d+136) then
        tmp = abs((t_1 + ((ew * (0.5d0 / eh)) * (ew * (tan(t) * sin(t))))))
    else
        tmp = 1.0d0 / abs((1.0d0 / (t_2 + (t_1 / (ew / (eh / tan(t)))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.cos(t);
	double t_2 = ew * Math.sin(t);
	double tmp;
	if (ew <= -1.25e+191) {
		tmp = Math.abs(t_2);
	} else if (ew <= 1.35e+136) {
		tmp = Math.abs((t_1 + ((ew * (0.5 / eh)) * (ew * (Math.tan(t) * Math.sin(t))))));
	} else {
		tmp = 1.0 / Math.abs((1.0 / (t_2 + (t_1 / (ew / (eh / Math.tan(t)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.cos(t)
	t_2 = ew * math.sin(t)
	tmp = 0
	if ew <= -1.25e+191:
		tmp = math.fabs(t_2)
	elif ew <= 1.35e+136:
		tmp = math.fabs((t_1 + ((ew * (0.5 / eh)) * (ew * (math.tan(t) * math.sin(t))))))
	else:
		tmp = 1.0 / math.fabs((1.0 / (t_2 + (t_1 / (ew / (eh / math.tan(t)))))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	t_2 = Float64(ew * sin(t))
	tmp = 0.0
	if (ew <= -1.25e+191)
		tmp = abs(t_2);
	elseif (ew <= 1.35e+136)
		tmp = abs(Float64(t_1 + Float64(Float64(ew * Float64(0.5 / eh)) * Float64(ew * Float64(tan(t) * sin(t))))));
	else
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(t_2 + Float64(t_1 / Float64(ew / Float64(eh / tan(t))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * cos(t);
	t_2 = ew * sin(t);
	tmp = 0.0;
	if (ew <= -1.25e+191)
		tmp = abs(t_2);
	elseif (ew <= 1.35e+136)
		tmp = abs((t_1 + ((ew * (0.5 / eh)) * (ew * (tan(t) * sin(t))))));
	else
		tmp = 1.0 / abs((1.0 / (t_2 + (t_1 / (ew / (eh / tan(t)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -1.25e+191], N[Abs[t$95$2], $MachinePrecision], If[LessEqual[ew, 1.35e+136], N[Abs[N[(t$95$1 + N[(N[(ew * N[(0.5 / eh), $MachinePrecision]), $MachinePrecision] * N[(ew * N[(N[Tan[t], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Abs[N[(1.0 / N[(t$95$2 + N[(t$95$1 / N[(ew / N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.25000000000000005e191

   1. Initial program 99.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. fma-define N/A

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

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

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in ew around inf

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

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

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

      2. sin-lowering-sin.f64 82.6%

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

   7. Simplified 82.6%

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

   if -1.25000000000000005e191 < ew < 1.3500000000000001e136

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. sin-atan N/A

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

      3. div-inv N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. fma-define N/A

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

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

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in ew around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

      15. cos-lowering-cos.f64 75.0%

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

   7. Simplified 75.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. tan-quot N/A

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

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

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

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

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

      15. tan-lowering-tan.f64 77.8%

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

   9. Applied egg-rr 77.8%

      \[\leadsto \left|eh \cdot \cos t + \color{blue}{\left(ew \cdot \frac{0.5}{eh}\right) \cdot \left(\left(\sin t \cdot \tan t\right) \cdot ew\right)}\right| \]

   if 1.3500000000000001e136 < ew

   1. Initial program 99.7%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. cos-atan N/A

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

      5. un-div-inv N/A

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

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

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

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

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

      8. hypot-1-def N/A

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

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

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

      10. associate-/l/ N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. tan-lowering-tan.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

   6. Applied egg-rr 86.0%

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

   7. Taylor expanded in eh around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{cos.f64}\left(t\right)\right), \mathsf{/.f64}\left(ew, \mathsf{/.f64}\left(eh, \mathsf{tan.f64}\left(t\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(t\right), ew\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 74.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.25 \cdot 10^{+191}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{elif}\;ew \leq 1.35 \cdot 10^{+136}:\\ \;\;\;\;\left|eh \cdot \cos t + \left(ew \cdot \frac{0.5}{eh}\right) \cdot \left(ew \cdot \left(\tan t \cdot \sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew \cdot \sin t + \frac{eh \cdot \cos t}{\frac{ew}{\frac{eh}{\tan t}}}}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;ew \leq -6.2 \cdot 10^{+190}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.85 \cdot 10^{+136}:\\ \;\;\;\;\left|eh \cdot \cos t + \left(ew \cdot \frac{0.5}{eh}\right) \cdot \left(ew \cdot \left(\tan t \cdot \sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= ew -6.2e+190)
     t_1
     (if (<= ew 1.85e+136)
       (fabs
        (+ (* eh (cos t)) (* (* ew (/ 0.5 eh)) (* ew (* (tan t) (sin t))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (ew <= -6.2e+190) {
		tmp = t_1;
	} else if (ew <= 1.85e+136) {
		tmp = fabs(((eh * cos(t)) + ((ew * (0.5 / eh)) * (ew * (tan(t) * 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 * sin(t)))
    if (ew <= (-6.2d+190)) then
        tmp = t_1
    else if (ew <= 1.85d+136) then
        tmp = abs(((eh * cos(t)) + ((ew * (0.5d0 / eh)) * (ew * (tan(t) * 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.sin(t)));
	double tmp;
	if (ew <= -6.2e+190) {
		tmp = t_1;
	} else if (ew <= 1.85e+136) {
		tmp = Math.abs(((eh * Math.cos(t)) + ((ew * (0.5 / eh)) * (ew * (Math.tan(t) * Math.sin(t))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.sin(t)))
	tmp = 0
	if ew <= -6.2e+190:
		tmp = t_1
	elif ew <= 1.85e+136:
		tmp = math.fabs(((eh * math.cos(t)) + ((ew * (0.5 / eh)) * (ew * (math.tan(t) * math.sin(t))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (ew <= -6.2e+190)
		tmp = t_1;
	elseif (ew <= 1.85e+136)
		tmp = abs(Float64(Float64(eh * cos(t)) + Float64(Float64(ew * Float64(0.5 / eh)) * Float64(ew * Float64(tan(t) * sin(t))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * sin(t)));
	tmp = 0.0;
	if (ew <= -6.2e+190)
		tmp = t_1;
	elseif (ew <= 1.85e+136)
		tmp = abs(((eh * cos(t)) + ((ew * (0.5 / eh)) * (ew * (tan(t) * 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[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -6.2e+190], t$95$1, If[LessEqual[ew, 1.85e+136], N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[(0.5 / eh), $MachinePrecision]), $MachinePrecision] * N[(ew * N[(N[Tan[t], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -6.2000000000000003e190 or 1.85000000000000005e136 < ew

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. fma-define N/A

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

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

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in ew around inf

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

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

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

      2. sin-lowering-sin.f64 77.8%

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

   7. Simplified 77.8%

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

   if -6.2000000000000003e190 < ew < 1.85000000000000005e136

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. sin-atan N/A

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

      3. div-inv N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. fma-define N/A

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

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

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in ew around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

      15. cos-lowering-cos.f64 75.0%

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

   7. Simplified 75.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

      12. tan-quot N/A

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

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

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

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

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

      15. tan-lowering-tan.f64 77.8%

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

   9. Applied egg-rr 77.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -6.2 \cdot 10^{+190}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{elif}\;ew \leq 1.85 \cdot 10^{+136}:\\ \;\;\;\;\left|eh \cdot \cos t + \left(ew \cdot \frac{0.5}{eh}\right) \cdot \left(ew \cdot \left(\tan t \cdot \sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;ew \leq -6.2 \cdot 10^{+190}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 6.2 \cdot 10^{+136}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= ew -6.2e+190) t_1 (if (<= ew 6.2e+136) (fabs (* eh (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (ew <= -6.2e+190) {
		tmp = t_1;
	} else if (ew <= 6.2e+136) {
		tmp = fabs((eh * cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((ew * sin(t)))
    if (ew <= (-6.2d+190)) then
        tmp = t_1
    else if (ew <= 6.2d+136) then
        tmp = abs((eh * cos(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * Math.sin(t)));
	double tmp;
	if (ew <= -6.2e+190) {
		tmp = t_1;
	} else if (ew <= 6.2e+136) {
		tmp = Math.abs((eh * Math.cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.sin(t)))
	tmp = 0
	if ew <= -6.2e+190:
		tmp = t_1
	elif ew <= 6.2e+136:
		tmp = math.fabs((eh * math.cos(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (ew <= -6.2e+190)
		tmp = t_1;
	elseif (ew <= 6.2e+136)
		tmp = abs(Float64(eh * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * sin(t)));
	tmp = 0.0;
	if (ew <= -6.2e+190)
		tmp = t_1;
	elseif (ew <= 6.2e+136)
		tmp = abs((eh * cos(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -6.2e+190], t$95$1, If[LessEqual[ew, 6.2e+136], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -6.2000000000000003e190 or 6.19999999999999967e136 < ew

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. fma-define N/A

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

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

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in ew around inf

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

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

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

      2. sin-lowering-sin.f64 77.8%

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

   7. Simplified 77.8%

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

   if -6.2000000000000003e190 < ew < 6.19999999999999967e136

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. sin-atan N/A

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

      3. div-inv N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. fma-define N/A

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

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

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

   4. Applied egg-rr 57.7%

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

   5. Taylor expanded in eh around inf

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

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

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

      2. cos-lowering-cos.f64 77.1%

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

   7. Simplified 77.1%

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

Alternative 13: 61.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \left|eh \cdot \cos t\right| \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. sin-atan N/A

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

   3. div-inv N/A

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

   4. associate-/l* N/A

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

   5. associate-*r* N/A

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

   6. fma-define N/A

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

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

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

4. Applied egg-rr 67.5%

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

5. Taylor expanded in eh around inf

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

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

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

   2. cos-lowering-cos.f64 63.5%

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

7. Simplified 63.5%

   \[\leadsto \left|\color{blue}{eh \cdot \cos t}\right| \]
8. Add Preprocessing

Alternative 14: 41.8% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. sin-atan N/A

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

   3. div-inv N/A

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

   4. associate-/l* N/A

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

   5. associate-*r* N/A

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

   6. fma-define N/A

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

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

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

4. Applied egg-rr 67.5%

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

5. Taylor expanded in t around 0

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

7. Simplified 45.6%

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

Reproduce

herbie shell --seed 2024159 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))