Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 22.7s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / Math.tan(t);
	return Math.abs((((ew * Math.sin(t)) / Math.hypot(1.0, t_1)) + ((eh * Math.cos(t)) * Math.sin(Math.atan(t_1)))));
}

Python

def code(eh, ew, t):
	t_1 = (eh / ew) / math.tan(t)
	return math.fabs((((ew * math.sin(t)) / math.hypot(1.0, t_1)) + ((eh * math.cos(t)) * math.sin(math.atan(t_1)))))

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	t_1 = (eh / ew) / tan(t);
	tmp = abs((((ew * sin(t)) / hypot(1.0, t_1)) + ((eh * cos(t)) * sin(atan(t_1)))));
end

Wolfram

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

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

   2. un-div-inv N/A

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

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

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

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

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

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

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

   6. hypot-1-def N/A

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

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

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

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

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

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

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

   10. tan-lowering-tan.f64 99.8%

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

4. Applied egg-rr 99.8%

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

Alternative 2: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))) + ((ew * sin(t)) * cos(atan((eh / (ew * 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)) * sin(atan(((eh / ew) / tan(t))))) + ((ew * sin(t)) * cos(atan((eh / (ew * t)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 99.2%

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 3: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs(((ew * sin(t)) + ((eh * cos(t)) * sin(atan((eh / (ew * tan(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(((ew * sin(t)) + ((eh * cos(t)) * sin(atan((eh / (ew * tan(t))))))))
end function

Java

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. un-div-inv N/A

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

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

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

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

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

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

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

   6. hypot-1-def N/A

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

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

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

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

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

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

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

   10. tan-lowering-tan.f64 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in eh around 0

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

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

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. sin-lowering-sin.f64 99.0%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

Alternative 4: 86.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ t_2 := \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{if}\;eh \leq -2.4 \cdot 10^{+134}:\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{elif}\;eh \leq 4.8 \cdot 10^{+156}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot t\_2\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{t\_1 \cdot t\_2}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))) (t_2 (sin (atan (/ (/ eh ew) (tan t))))))
   (if (<= eh -2.4e+134)
     (fabs (* t_1 (sin (atan (/ eh (* ew (tan t)))))))
     (if (<= eh 4.8e+156)
       (fabs (+ (* ew (sin t)) (* eh t_2)))
       (/ 1.0 (fabs (/ 1.0 (* t_1 t_2))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double t_2 = sin(atan(((eh / ew) / tan(t))));
	double tmp;
	if (eh <= -2.4e+134) {
		tmp = fabs((t_1 * sin(atan((eh / (ew * tan(t)))))));
	} else if (eh <= 4.8e+156) {
		tmp = fabs(((ew * sin(t)) + (eh * t_2)));
	} else {
		tmp = 1.0 / fabs((1.0 / (t_1 * t_2)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = eh * cos(t)
    t_2 = sin(atan(((eh / ew) / tan(t))))
    if (eh <= (-2.4d+134)) then
        tmp = abs((t_1 * sin(atan((eh / (ew * tan(t)))))))
    else if (eh <= 4.8d+156) then
        tmp = abs(((ew * sin(t)) + (eh * t_2)))
    else
        tmp = 1.0d0 / abs((1.0d0 / (t_1 * t_2)))
    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 = Math.sin(Math.atan(((eh / ew) / Math.tan(t))));
	double tmp;
	if (eh <= -2.4e+134) {
		tmp = Math.abs((t_1 * Math.sin(Math.atan((eh / (ew * Math.tan(t)))))));
	} else if (eh <= 4.8e+156) {
		tmp = Math.abs(((ew * Math.sin(t)) + (eh * t_2)));
	} else {
		tmp = 1.0 / Math.abs((1.0 / (t_1 * t_2)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.cos(t)
	t_2 = math.sin(math.atan(((eh / ew) / math.tan(t))))
	tmp = 0
	if eh <= -2.4e+134:
		tmp = math.fabs((t_1 * math.sin(math.atan((eh / (ew * math.tan(t)))))))
	elif eh <= 4.8e+156:
		tmp = math.fabs(((ew * math.sin(t)) + (eh * t_2)))
	else:
		tmp = 1.0 / math.fabs((1.0 / (t_1 * t_2)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	t_2 = sin(atan(Float64(Float64(eh / ew) / tan(t))))
	tmp = 0.0
	if (eh <= -2.4e+134)
		tmp = abs(Float64(t_1 * sin(atan(Float64(eh / Float64(ew * tan(t)))))));
	elseif (eh <= 4.8e+156)
		tmp = abs(Float64(Float64(ew * sin(t)) + Float64(eh * t_2)));
	else
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(t_1 * t_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * cos(t);
	t_2 = sin(atan(((eh / ew) / tan(t))));
	tmp = 0.0;
	if (eh <= -2.4e+134)
		tmp = abs((t_1 * sin(atan((eh / (ew * tan(t)))))));
	elseif (eh <= 4.8e+156)
		tmp = abs(((ew * sin(t)) + (eh * t_2)));
	else
		tmp = 1.0 / abs((1.0 / (t_1 * t_2)));
	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[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.4e+134], N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 4.8e+156], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(eh * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Abs[N[(1.0 / N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -2.40000000000000005e134

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. cos-lowering-cos.f64 89.9%

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

   5. Simplified 89.9%

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

   if -2.40000000000000005e134 < eh < 4.8000000000000002e156

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

   5. Simplified 87.3%

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

      1. cos-atan N/A

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

      2. metadata-eval N/A

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

      3. div-inv N/A

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

      4. clear-num N/A

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

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

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

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

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

      7. hypot-undefine N/A

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

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

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

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

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

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

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

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

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

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

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

      13. sin-lowering-sin.f64 87.2%

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

   7. Applied egg-rr 87.2%

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

   8. Taylor expanded in eh around 0

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

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

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

      2. sin-lowering-sin.f64 87.2%

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

   10. Simplified 87.2%

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

   if 4.8000000000000002e156 < eh

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in ew around 0

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

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

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

      2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. associate-/r* N/A

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

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

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

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

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

      11. tan-lowering-tan.f64 96.3%

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

   7. Simplified 96.3%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.4 \cdot 10^{+134}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{elif}\;eh \leq 4.8 \cdot 10^{+156}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{if}\;eh \leq -1.72 \cdot 10^{+134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 5.2 \cdot 10^{+155}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (* eh (cos t)) (sin (atan (/ eh (* ew (tan t)))))))))
   (if (<= eh -1.72e+134)
     t_1
     (if (<= eh 5.2e+155)
       (fabs (+ (* ew (sin t)) (* eh (sin (atan (/ (/ eh ew) (tan t)))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))));
	double tmp;
	if (eh <= -1.72e+134) {
		tmp = t_1;
	} else if (eh <= 5.2e+155) {
		tmp = fabs(((ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(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(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))))
    if (eh <= (-1.72d+134)) then
        tmp = t_1
    else if (eh <= 5.2d+155) then
        tmp = abs(((ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(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(((eh * Math.cos(t)) * Math.sin(Math.atan((eh / (ew * Math.tan(t)))))));
	double tmp;
	if (eh <= -1.72e+134) {
		tmp = t_1;
	} else if (eh <= 5.2e+155) {
		tmp = Math.abs(((ew * Math.sin(t)) + (eh * Math.sin(Math.atan(((eh / ew) / Math.tan(t)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs(((eh * math.cos(t)) * math.sin(math.atan((eh / (ew * math.tan(t)))))))
	tmp = 0
	if eh <= -1.72e+134:
		tmp = t_1
	elif eh <= 5.2e+155:
		tmp = math.fabs(((ew * math.sin(t)) + (eh * math.sin(math.atan(((eh / ew) / math.tan(t)))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(eh * cos(t)) * sin(atan(Float64(eh / Float64(ew * tan(t)))))))
	tmp = 0.0
	if (eh <= -1.72e+134)
		tmp = t_1;
	elseif (eh <= 5.2e+155)
		tmp = abs(Float64(Float64(ew * sin(t)) + Float64(eh * sin(atan(Float64(Float64(eh / ew) / tan(t)))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))));
	tmp = 0.0;
	if (eh <= -1.72e+134)
		tmp = t_1;
	elseif (eh <= 5.2e+155)
		tmp = abs(((ew * sin(t)) + (eh * sin(atan(((eh / ew) / tan(t)))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.72e+134], t$95$1, If[LessEqual[eh, 5.2e+155], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.71999999999999998e134 or 5.2000000000000004e155 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. cos-lowering-cos.f64 92.9%

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

   5. Simplified 92.9%

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

   if -1.71999999999999998e134 < eh < 5.2000000000000004e155

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

   5. Simplified 87.3%

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

      1. cos-atan N/A

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

      2. metadata-eval N/A

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

      3. div-inv N/A

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

      4. clear-num N/A

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

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

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

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

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

      7. hypot-undefine N/A

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

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

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

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

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

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

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

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

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

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

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

      13. sin-lowering-sin.f64 87.2%

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

   7. Applied egg-rr 87.2%

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

   8. Taylor expanded in eh around 0

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

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

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

      2. sin-lowering-sin.f64 87.2%

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

   10. Simplified 87.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.72 \cdot 10^{+134}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{elif}\;eh \leq 5.2 \cdot 10^{+155}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{if}\;eh \leq -4.1 \cdot 10^{-37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 4.8 \cdot 10^{-18}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (* eh (cos t)) (sin (atan (/ eh (* ew (tan t)))))))))
   (if (<= eh -4.1e-37) t_1 (if (<= eh 4.8e-18) (fabs (* ew (sin t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))));
	double tmp;
	if (eh <= -4.1e-37) {
		tmp = t_1;
	} else if (eh <= 4.8e-18) {
		tmp = fabs((ew * 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(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))))
    if (eh <= (-4.1d-37)) then
        tmp = t_1
    else if (eh <= 4.8d-18) then
        tmp = abs((ew * 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(((eh * Math.cos(t)) * Math.sin(Math.atan((eh / (ew * Math.tan(t)))))));
	double tmp;
	if (eh <= -4.1e-37) {
		tmp = t_1;
	} else if (eh <= 4.8e-18) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs(((eh * math.cos(t)) * math.sin(math.atan((eh / (ew * math.tan(t)))))))
	tmp = 0
	if eh <= -4.1e-37:
		tmp = t_1
	elif eh <= 4.8e-18:
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(eh * cos(t)) * sin(atan(Float64(eh / Float64(ew * tan(t)))))))
	tmp = 0.0
	if (eh <= -4.1e-37)
		tmp = t_1;
	elseif (eh <= 4.8e-18)
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))));
	tmp = 0.0;
	if (eh <= -4.1e-37)
		tmp = t_1;
	elseif (eh <= 4.8e-18)
		tmp = abs((ew * sin(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -4.1e-37], t$95$1, If[LessEqual[eh, 4.8e-18], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.0999999999999998e-37 or 4.79999999999999988e-18 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. cos-lowering-cos.f64 77.9%

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

   5. Simplified 77.9%

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

   if -4.0999999999999998e-37 < eh < 4.79999999999999988e-18

   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. cos-atan N/A

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

      2. un-div-inv N/A

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

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

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

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

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

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

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

      6. hypot-1-def N/A

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

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

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

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

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

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

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

      10. tan-lowering-tan.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   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 79.0%

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

   7. Simplified 79.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.1 \cdot 10^{-37}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{elif}\;eh \leq 4.8 \cdot 10^{-18}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.36 \cdot 10^{+134}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{elif}\;eh \leq 2.05 \cdot 10^{-18}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -1.36e+134)
   (fabs eh)
   (if (<= eh 2.05e-18) (fabs (* ew (sin t))) (fabs eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.36e+134) {
		tmp = fabs(eh);
	} else if (eh <= 2.05e-18) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs(eh);
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= (-1.36d+134)) then
        tmp = abs(eh)
    else if (eh <= 2.05d-18) then
        tmp = abs((ew * sin(t)))
    else
        tmp = abs(eh)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.36e+134) {
		tmp = Math.abs(eh);
	} else if (eh <= 2.05e-18) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = Math.abs(eh);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -1.36e+134:
		tmp = math.fabs(eh)
	elif eh <= 2.05e-18:
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs(eh)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -1.36e+134)
		tmp = abs(eh);
	elseif (eh <= 2.05e-18)
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(eh);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -1.36e+134)
		tmp = abs(eh);
	elseif (eh <= 2.05e-18)
		tmp = abs((ew * sin(t)));
	else
		tmp = abs(eh);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -1.36e+134], N[Abs[eh], $MachinePrecision], If[LessEqual[eh, 2.05e-18], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[eh], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.35999999999999995e134 or 2.0499999999999999e-18 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

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

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

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

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

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

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

      6. tan-lowering-tan.f64 52.8%

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

   5. Simplified 52.8%

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

      1. associate-/r* N/A

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

      2. sin-atan N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. hypot-undefine N/A

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

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

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

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

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

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

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

      15. tan-lowering-tan.f64 25.9%

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

   7. Applied egg-rr 25.9%

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

   8. Taylor expanded in eh around inf

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

   10. Simplified 53.1%

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

   if -1.35999999999999995e134 < eh < 2.0499999999999999e-18

   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. cos-atan N/A

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

      2. un-div-inv N/A

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

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

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

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

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

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

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

      6. hypot-1-def N/A

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

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

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

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

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

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

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

      10. tan-lowering-tan.f64 99.8%

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

   4. Applied egg-rr 99.8%

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

   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 70.9%

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

   7. Simplified 70.9%

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

Alternative 8: 43.1% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -5.8 \cdot 10^{-42}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{elif}\;eh \leq 2.3 \cdot 10^{-44}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -5.8e-42)
   (fabs eh)
   (if (<= eh 2.3e-44) (fabs (* ew t)) (fabs eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -5.8e-42) {
		tmp = fabs(eh);
	} else if (eh <= 2.3e-44) {
		tmp = fabs((ew * t));
	} else {
		tmp = fabs(eh);
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= (-5.8d-42)) then
        tmp = abs(eh)
    else if (eh <= 2.3d-44) then
        tmp = abs((ew * t))
    else
        tmp = abs(eh)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -5.8e-42) {
		tmp = Math.abs(eh);
	} else if (eh <= 2.3e-44) {
		tmp = Math.abs((ew * t));
	} else {
		tmp = Math.abs(eh);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -5.8e-42:
		tmp = math.fabs(eh)
	elif eh <= 2.3e-44:
		tmp = math.fabs((ew * t))
	else:
		tmp = math.fabs(eh)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -5.8e-42)
		tmp = abs(eh);
	elseif (eh <= 2.3e-44)
		tmp = abs(Float64(ew * t));
	else
		tmp = abs(eh);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -5.8e-42)
		tmp = abs(eh);
	elseif (eh <= 2.3e-44)
		tmp = abs((ew * t));
	else
		tmp = abs(eh);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -5.8e-42], N[Abs[eh], $MachinePrecision], If[LessEqual[eh, 2.3e-44], N[Abs[N[(ew * t), $MachinePrecision]], $MachinePrecision], N[Abs[eh], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.8000000000000006e-42 or 2.29999999999999998e-44 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

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

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

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

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

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

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

      6. tan-lowering-tan.f64 46.2%

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

   5. Simplified 46.2%

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

      1. associate-/r* N/A

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

      2. sin-atan N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. hypot-undefine N/A

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

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

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

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

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

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

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

      15. tan-lowering-tan.f64 25.8%

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

   7. Applied egg-rr 25.8%

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

   8. Taylor expanded in eh around inf

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

   10. Simplified 46.6%

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

   if -5.8000000000000006e-42 < eh < 2.29999999999999998e-44

   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. cos-atan N/A

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

      2. un-div-inv N/A

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

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

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

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

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

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

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

      6. hypot-1-def N/A

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

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

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

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

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

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

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

      10. tan-lowering-tan.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   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 80.8%

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

   7. Simplified 80.8%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 44.8%

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

   10. Simplified 44.8%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -5.8 \cdot 10^{-42}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{elif}\;eh \leq 2.3 \cdot 10^{-44}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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. Taylor expanded in t around 0

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

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

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

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

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

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

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

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

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

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

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

   6. tan-lowering-tan.f64 32.4%

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

5. Simplified 32.4%

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

   1. associate-/r* N/A

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

   2. sin-atan N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

   11. hypot-undefine N/A

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

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

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

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

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

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

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

   15. tan-lowering-tan.f64 20.7%

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

7. Applied egg-rr 20.7%

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

8. Taylor expanded in eh around inf

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

10. Simplified 33.0%

    \[\leadsto \left|\color{blue}{eh}\right| \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(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))))))))