Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.7s

Alternatives: 8

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = eh * tan(t)
    code = abs((((eh * sin(t)) * sin(atan((t_1 / (0.0d0 - ew))))) - ((ew * cos(atan((t_1 / ew)))) * cos(t))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 98.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 66.1%

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

6. Applied egg-rr 82.2%

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

7. Taylor expanded in ew around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

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

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

   4. mul-1-neg N/A

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

   5. neg-sub0 N/A

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

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

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

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

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

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

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

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

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

9. Simplified 98.2%

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

10. Taylor expanded in eh around 0

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

    1. fabs-neg N/A

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

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

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

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

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

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

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

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

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

    6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

    12. associate-/l* N/A

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

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

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

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

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

    15. tan-lowering-tan.f64 98.2

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

12. Simplified 98.2%

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

Alternative 3: 74.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;eh \leq -1.45 \cdot 10^{+162}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 6.5 \cdot 10^{+72}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right) \cdot \left(ew \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= eh -1.45e+162)
     t_1
     (if (<= eh 6.5e+72)
       (fabs (* (cos (atan (/ (* eh (tan t)) ew))) (* ew (cos t))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (eh <= -1.45e+162) {
		tmp = t_1;
	} else if (eh <= 6.5e+72) {
		tmp = fabs((cos(atan(((eh * tan(t)) / ew))) * (ew * cos(t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((eh * sin(t)))
    if (eh <= (-1.45d+162)) then
        tmp = t_1
    else if (eh <= 6.5d+72) then
        tmp = abs((cos(atan(((eh * tan(t)) / ew))) * (ew * cos(t))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * Math.sin(t)));
	double tmp;
	if (eh <= -1.45e+162) {
		tmp = t_1;
	} else if (eh <= 6.5e+72) {
		tmp = Math.abs((Math.cos(Math.atan(((eh * Math.tan(t)) / ew))) * (ew * Math.cos(t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	tmp = 0
	if eh <= -1.45e+162:
		tmp = t_1
	elif eh <= 6.5e+72:
		tmp = math.fabs((math.cos(math.atan(((eh * math.tan(t)) / ew))) * (ew * math.cos(t))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (eh <= -1.45e+162)
		tmp = t_1;
	elseif (eh <= 6.5e+72)
		tmp = abs(Float64(cos(atan(Float64(Float64(eh * tan(t)) / ew))) * Float64(ew * cos(t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	tmp = 0.0;
	if (eh <= -1.45e+162)
		tmp = t_1;
	elseif (eh <= 6.5e+72)
		tmp = abs((cos(atan(((eh * tan(t)) / ew))) * (ew * cos(t))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.45e+162], t$95$1, If[LessEqual[eh, 6.5e+72], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.45000000000000003e162 or 6.5000000000000001e72 < eh

   1. Initial program 99.8%

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

   4. Applied egg-rr 31.2%

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

   6. Applied egg-rr 51.5%

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

   7. Taylor expanded in eh around -inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   8. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 76.6

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

   9. Simplified 76.6%

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

   if -1.45000000000000003e162 < eh < 6.5000000000000001e72

   1. Initial program 99.9%

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

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

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in ew around inf

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. tan-lowering-tan.f64 77.0

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

   7. Simplified 77.0%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.45 \cdot 10^{+162}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{elif}\;eh \leq 6.5 \cdot 10^{+72}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right) \cdot \left(ew \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;t \leq -0.00066:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1480:\\ \;\;\;\;\left|eh \cdot t + ew \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= t -0.00066)
     t_1
     (if (<= t 1480.0)
       (fabs (+ (* eh t) (* ew (cos (atan (* eh (/ (tan t) ew)))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (t <= -0.00066) {
		tmp = t_1;
	} else if (t <= 1480.0) {
		tmp = fabs(((eh * t) + (ew * cos(atan((eh * (tan(t) / ew)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((eh * sin(t)))
    if (t <= (-0.00066d0)) then
        tmp = t_1
    else if (t <= 1480.0d0) then
        tmp = abs(((eh * t) + (ew * cos(atan((eh * (tan(t) / ew)))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * Math.sin(t)));
	double tmp;
	if (t <= -0.00066) {
		tmp = t_1;
	} else if (t <= 1480.0) {
		tmp = Math.abs(((eh * t) + (ew * Math.cos(Math.atan((eh * (Math.tan(t) / ew)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	tmp = 0
	if t <= -0.00066:
		tmp = t_1
	elif t <= 1480.0:
		tmp = math.fabs(((eh * t) + (ew * math.cos(math.atan((eh * (math.tan(t) / ew)))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (t <= -0.00066)
		tmp = t_1;
	elseif (t <= 1480.0)
		tmp = abs(Float64(Float64(eh * t) + Float64(ew * cos(atan(Float64(eh * Float64(tan(t) / ew)))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	tmp = 0.0;
	if (t <= -0.00066)
		tmp = t_1;
	elseif (t <= 1480.0)
		tmp = abs(((eh * t) + (ew * cos(atan((eh * (tan(t) / ew)))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -0.00066], t$95$1, If[LessEqual[t, 1480.0], N[Abs[N[(N[(eh * t), $MachinePrecision] + N[(ew * N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.6e-4 or 1480 < t

   1. Initial program 99.7%

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

   4. Applied egg-rr 58.4%

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

   6. Applied egg-rr 75.4%

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

   7. Taylor expanded in eh around -inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   8. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 49.5

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

   9. Simplified 49.5%

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

   if -6.6e-4 < t < 1480

   1. Initial program 100.0%

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

   4. Applied egg-rr 73.3%

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

   6. Applied egg-rr 88.7%

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

   7. Taylor expanded in ew around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-sub0 N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

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

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

   9. Simplified 98.8%

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

   10. Taylor expanded in t around 0

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

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

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

       2. mul-1-neg N/A

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

       3. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

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

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

       9. associate-/l* N/A

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

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

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

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

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

       12. tan-lowering-tan.f64 98.0

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

   12. Simplified 98.0%

       \[\leadsto \left|\color{blue}{\left(0 - eh \cdot t\right) - ew \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00066:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{elif}\;t \leq 1480:\\ \;\;\;\;\left|eh \cdot t + ew \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.4% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;t \leq -8 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-50}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= t -8e-9) t_1 (if (<= t 6.8e-50) (fabs ew) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (t <= -8e-9) {
		tmp = t_1;
	} else if (t <= 6.8e-50) {
		tmp = fabs(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((eh * sin(t)))
    if (t <= (-8d-9)) then
        tmp = t_1
    else if (t <= 6.8d-50) then
        tmp = abs(ew)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * Math.sin(t)));
	double tmp;
	if (t <= -8e-9) {
		tmp = t_1;
	} else if (t <= 6.8e-50) {
		tmp = Math.abs(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	tmp = 0
	if t <= -8e-9:
		tmp = t_1
	elif t <= 6.8e-50:
		tmp = math.fabs(ew)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (t <= -8e-9)
		tmp = t_1;
	elseif (t <= 6.8e-50)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	tmp = 0.0;
	if (t <= -8e-9)
		tmp = t_1;
	elseif (t <= 6.8e-50)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -8e-9], t$95$1, If[LessEqual[t, 6.8e-50], N[Abs[ew], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.0000000000000005e-9 or 6.80000000000000029e-50 < t

   1. Initial program 99.7%

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

   4. Applied egg-rr 57.3%

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

   6. Applied egg-rr 73.6%

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

   7. Taylor expanded in eh around -inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   8. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 50.4

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

   9. Simplified 50.4%

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

   if -8.0000000000000005e-9 < t < 6.80000000000000029e-50

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

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

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

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

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

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

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

      10. mul-1-neg N/A

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

      11. neg-sub0 N/A

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

      12. --lowering--.f64 72.9

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

   5. Simplified 72.9%

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

      1. flip3-- N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{\color{blue}{\frac{{0}^{3} - {ew}^{3}}{0 \cdot 0 + \left(ew \cdot ew + 0 \cdot ew\right)}}}\right)\right| \]

      2. associate-/r/ N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(0 \cdot 0 + \left(ew \cdot ew + 0 \cdot ew\right)\right)\right)}\right| \]

      3. metadata-eval N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(\color{blue}{0} + \left(ew \cdot ew + 0 \cdot ew\right)\right)\right)\right| \]

      4. +-lft-identity N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \color{blue}{\left(ew \cdot ew + 0 \cdot ew\right)}\right)\right| \]

      5. distribute-rgt-out N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \color{blue}{\left(ew \cdot \left(ew + 0\right)\right)}\right)\right| \]

      6. +-commutative N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot \color{blue}{\left(0 + ew\right)}\right)\right)\right| \]

      7. +-lft-identity N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot \color{blue}{ew}\right)\right)\right| \]

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

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot ew\right)\right)}\right| \]

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

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\color{blue}{\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}}} \cdot \left(ew \cdot ew\right)\right)\right| \]

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

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{\color{blue}{eh \cdot \tan t}}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot ew\right)\right)\right| \]

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

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \color{blue}{\tan t}}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot ew\right)\right)\right| \]

      12. metadata-eval N/A

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{\color{blue}{0} - {ew}^{3}} \cdot \left(ew \cdot ew\right)\right)\right| \]

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

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{\color{blue}{0 - {ew}^{3}}} \cdot \left(ew \cdot ew\right)\right)\right| \]

      14. cube-mult N/A

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

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

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

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

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

      17. *-lowering-*.f64 25.2

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

   7. Applied egg-rr 25.2%

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

   9. Applied egg-rr 72.7%

      \[\leadsto \left|ew \cdot \color{blue}{{\left(1 + {\left(\frac{\frac{ew}{\tan t}}{eh}\right)}^{-2}\right)}^{-0.5}}\right| \]

   10. Taylor expanded in ew around inf

       \[\leadsto \left|\color{blue}{ew}\right| \]
   11. Step-by-step derivation

   12. Simplified 73.1%

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

Alternative 6: 46.1% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.05 \cdot 10^{+163}:\\ \;\;\;\;eh \cdot \sin t\\ \mathbf{elif}\;eh \leq 2.5 \cdot 10^{+162}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -1.05e+163)
   (* eh (sin t))
   (if (<= eh 2.5e+162) (fabs ew) (fabs (* eh t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.05e+163) {
		tmp = eh * sin(t);
	} else if (eh <= 2.5e+162) {
		tmp = fabs(ew);
	} else {
		tmp = fabs((eh * t));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= (-1.05d+163)) then
        tmp = eh * sin(t)
    else if (eh <= 2.5d+162) then
        tmp = abs(ew)
    else
        tmp = abs((eh * t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.05e+163) {
		tmp = eh * Math.sin(t);
	} else if (eh <= 2.5e+162) {
		tmp = Math.abs(ew);
	} else {
		tmp = Math.abs((eh * t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -1.05e+163:
		tmp = eh * math.sin(t)
	elif eh <= 2.5e+162:
		tmp = math.fabs(ew)
	else:
		tmp = math.fabs((eh * t))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -1.05e+163)
		tmp = Float64(eh * sin(t));
	elseif (eh <= 2.5e+162)
		tmp = abs(ew);
	else
		tmp = abs(Float64(eh * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -1.05e+163)
		tmp = eh * sin(t);
	elseif (eh <= 2.5e+162)
		tmp = abs(ew);
	else
		tmp = abs((eh * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -1.05e+163], N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[eh, 2.5e+162], N[Abs[ew], $MachinePrecision], N[Abs[N[(eh * t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eh < -1.05e163

   1. Initial program 99.8%

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

   4. Applied egg-rr 28.8%

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

   6. Applied egg-rr 37.0%

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

   7. Taylor expanded in eh around -inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   8. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 79.8

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

   9. Simplified 79.8%

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   10. Step-by-step derivation

       1. neg-fabs N/A

          \[\leadsto \color{blue}{\left|\mathsf{neg}\left(eh \cdot \sin t\right)\right|} \]

       2. sub0-neg N/A

          \[\leadsto \left|\color{blue}{0 - eh \cdot \sin t}\right| \]

       3. flip3-- N/A

          \[\leadsto \left|\color{blue}{\frac{{0}^{3} - {\left(eh \cdot \sin t\right)}^{3}}{0 \cdot 0 + \left(\left(eh \cdot \sin t\right) \cdot \left(eh \cdot \sin t\right) + 0 \cdot \left(eh \cdot \sin t\right)\right)}}\right| \]

       4. fabs-div N/A

          \[\leadsto \color{blue}{\frac{\left|{0}^{3} - {\left(eh \cdot \sin t\right)}^{3}\right|}{\left|0 \cdot 0 + \left(\left(eh \cdot \sin t\right) \cdot \left(eh \cdot \sin t\right) + 0 \cdot \left(eh \cdot \sin t\right)\right)\right|}} \]

   11. Applied egg-rr 50.9%

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

   if -1.05e163 < eh < 2.4999999999999998e162

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

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

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

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

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

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

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

      10. mul-1-neg N/A

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

      11. neg-sub0 N/A

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

      12. --lowering--.f64 49.9

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

   5. Simplified 49.9%

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

      1. flip3-- N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{\color{blue}{\frac{{0}^{3} - {ew}^{3}}{0 \cdot 0 + \left(ew \cdot ew + 0 \cdot ew\right)}}}\right)\right| \]

      2. associate-/r/ N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(0 \cdot 0 + \left(ew \cdot ew + 0 \cdot ew\right)\right)\right)}\right| \]

      3. metadata-eval N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(\color{blue}{0} + \left(ew \cdot ew + 0 \cdot ew\right)\right)\right)\right| \]

      4. +-lft-identity N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \color{blue}{\left(ew \cdot ew + 0 \cdot ew\right)}\right)\right| \]

      5. distribute-rgt-out N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \color{blue}{\left(ew \cdot \left(ew + 0\right)\right)}\right)\right| \]

      6. +-commutative N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot \color{blue}{\left(0 + ew\right)}\right)\right)\right| \]

      7. +-lft-identity N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot \color{blue}{ew}\right)\right)\right| \]

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

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot ew\right)\right)}\right| \]

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

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\color{blue}{\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}}} \cdot \left(ew \cdot ew\right)\right)\right| \]

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

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{\color{blue}{eh \cdot \tan t}}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot ew\right)\right)\right| \]

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

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \color{blue}{\tan t}}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot ew\right)\right)\right| \]

      12. metadata-eval N/A

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{\color{blue}{0} - {ew}^{3}} \cdot \left(ew \cdot ew\right)\right)\right| \]

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

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{\color{blue}{0 - {ew}^{3}}} \cdot \left(ew \cdot ew\right)\right)\right| \]

      14. cube-mult N/A

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

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

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

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

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

      17. *-lowering-*.f64 19.7

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

   7. Applied egg-rr 19.7%

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

   9. Applied egg-rr 49.7%

      \[\leadsto \left|ew \cdot \color{blue}{{\left(1 + {\left(\frac{\frac{ew}{\tan t}}{eh}\right)}^{-2}\right)}^{-0.5}}\right| \]

   10. Taylor expanded in ew around inf

       \[\leadsto \left|\color{blue}{ew}\right| \]
   11. Step-by-step derivation

   12. Simplified 50.0%

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

   if 2.4999999999999998e162 < eh

   1. Initial program 99.8%

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

   4. Applied egg-rr 24.9%

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

   6. Applied egg-rr 56.9%

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

   7. Taylor expanded in eh around -inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   8. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 83.6

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

   9. Simplified 83.6%

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

   10. Taylor expanded in t around 0

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

       1. *-lowering-*.f64 58.0

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

   12. Simplified 58.0%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.05 \cdot 10^{+163}:\\ \;\;\;\;eh \cdot \sin t\\ \mathbf{elif}\;eh \leq 2.5 \cdot 10^{+162}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.3% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq 1.35 \cdot 10^{+161}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh 1.35e+161) (fabs ew) (fabs (* eh t))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 1.35e+161) {
		tmp = fabs(ew);
	} else {
		tmp = fabs((eh * t));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= 1.35d+161) then
        tmp = abs(ew)
    else
        tmp = abs((eh * t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 1.35e+161) {
		tmp = Math.abs(ew);
	} else {
		tmp = Math.abs((eh * t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= 1.35e+161:
		tmp = math.fabs(ew)
	else:
		tmp = math.fabs((eh * t))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= 1.35e+161)
		tmp = abs(ew);
	else
		tmp = abs(Float64(eh * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= 1.35e+161)
		tmp = abs(ew);
	else
		tmp = abs((eh * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, 1.35e+161], N[Abs[ew], $MachinePrecision], N[Abs[N[(eh * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 1.3499999999999999e161

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

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

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

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

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

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

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

      10. mul-1-neg N/A

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

      11. neg-sub0 N/A

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

      12. --lowering--.f64 46.3

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

   5. Simplified 46.3%

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

      1. flip3-- N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{\color{blue}{\frac{{0}^{3} - {ew}^{3}}{0 \cdot 0 + \left(ew \cdot ew + 0 \cdot ew\right)}}}\right)\right| \]

      2. associate-/r/ N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(0 \cdot 0 + \left(ew \cdot ew + 0 \cdot ew\right)\right)\right)}\right| \]

      3. metadata-eval N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(\color{blue}{0} + \left(ew \cdot ew + 0 \cdot ew\right)\right)\right)\right| \]

      4. +-lft-identity N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \color{blue}{\left(ew \cdot ew + 0 \cdot ew\right)}\right)\right| \]

      5. distribute-rgt-out N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \color{blue}{\left(ew \cdot \left(ew + 0\right)\right)}\right)\right| \]

      6. +-commutative N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot \color{blue}{\left(0 + ew\right)}\right)\right)\right| \]

      7. +-lft-identity N/A

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot \color{blue}{ew}\right)\right)\right| \]

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

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot ew\right)\right)}\right| \]

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

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\color{blue}{\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}}} \cdot \left(ew \cdot ew\right)\right)\right| \]

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

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{\color{blue}{eh \cdot \tan t}}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot ew\right)\right)\right| \]

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

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \color{blue}{\tan t}}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot ew\right)\right)\right| \]

      12. metadata-eval N/A

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{\color{blue}{0} - {ew}^{3}} \cdot \left(ew \cdot ew\right)\right)\right| \]

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

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{\color{blue}{0 - {ew}^{3}}} \cdot \left(ew \cdot ew\right)\right)\right| \]

      14. cube-mult N/A

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

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

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

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

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

      17. *-lowering-*.f64 18.0

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

   7. Applied egg-rr 18.0%

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

   9. Applied egg-rr 46.0%

      \[\leadsto \left|ew \cdot \color{blue}{{\left(1 + {\left(\frac{\frac{ew}{\tan t}}{eh}\right)}^{-2}\right)}^{-0.5}}\right| \]

   10. Taylor expanded in ew around inf

       \[\leadsto \left|\color{blue}{ew}\right| \]
   11. Step-by-step derivation

   12. Simplified 46.5%

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

   if 1.3499999999999999e161 < eh

   1. Initial program 99.8%

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

   4. Applied egg-rr 24.9%

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

   6. Applied egg-rr 56.9%

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

   7. Taylor expanded in eh around -inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   8. Step-by-step derivation

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

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

      2. sin-lowering-sin.f64 83.6

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

   9. Simplified 83.6%

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

   10. Taylor expanded in t around 0

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

       1. *-lowering-*.f64 58.0

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

   12. Simplified 58.0%

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

Alternative 8: 43.0% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

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

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

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

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

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

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

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

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

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

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

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

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

   10. mul-1-neg N/A

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

   11. neg-sub0 N/A

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

   12. --lowering--.f64 42.3

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

5. Simplified 42.3%

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

   1. flip3-- N/A

      \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{\color{blue}{\frac{{0}^{3} - {ew}^{3}}{0 \cdot 0 + \left(ew \cdot ew + 0 \cdot ew\right)}}}\right)\right| \]

   2. associate-/r/ N/A

      \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(0 \cdot 0 + \left(ew \cdot ew + 0 \cdot ew\right)\right)\right)}\right| \]

   3. metadata-eval N/A

      \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(\color{blue}{0} + \left(ew \cdot ew + 0 \cdot ew\right)\right)\right)\right| \]

   4. +-lft-identity N/A

      \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \color{blue}{\left(ew \cdot ew + 0 \cdot ew\right)}\right)\right| \]

   5. distribute-rgt-out N/A

      \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \color{blue}{\left(ew \cdot \left(ew + 0\right)\right)}\right)\right| \]

   6. +-commutative N/A

      \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot \color{blue}{\left(0 + ew\right)}\right)\right)\right| \]

   7. +-lft-identity N/A

      \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot \color{blue}{ew}\right)\right)\right| \]

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

      \[\leadsto \left|ew \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot ew\right)\right)}\right| \]

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

      \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\color{blue}{\frac{eh \cdot \tan t}{{0}^{3} - {ew}^{3}}} \cdot \left(ew \cdot ew\right)\right)\right| \]

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

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{\color{blue}{eh \cdot \tan t}}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot ew\right)\right)\right| \]

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

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \color{blue}{\tan t}}{{0}^{3} - {ew}^{3}} \cdot \left(ew \cdot ew\right)\right)\right| \]

   12. metadata-eval N/A

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{\color{blue}{0} - {ew}^{3}} \cdot \left(ew \cdot ew\right)\right)\right| \]

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

       \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{\color{blue}{0 - {ew}^{3}}} \cdot \left(ew \cdot ew\right)\right)\right| \]

   14. cube-mult N/A

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

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

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

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

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

   17. *-lowering-*.f64 16.1

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

7. Applied egg-rr 16.1%

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

9. Applied egg-rr 42.0%

   \[\leadsto \left|ew \cdot \color{blue}{{\left(1 + {\left(\frac{\frac{ew}{\tan t}}{eh}\right)}^{-2}\right)}^{-0.5}}\right| \]

10. Taylor expanded in ew around inf

    \[\leadsto \left|\color{blue}{ew}\right| \]
11. Step-by-step derivation

12. Simplified 42.5%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))