Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 16.2s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew + N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t / ew), $MachinePrecision] * eh), $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 rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 99.2

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

7. Applied rewrites 99.2%

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

8. Final simplification 99.2%

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

Alternative 3: 90.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot ew\\ t_2 := \left|\mathsf{fma}\left(\left(-eh\right) \cdot \sin t, \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right), \cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right) \cdot t\_1\right)\right|\\ \mathbf{if}\;eh \leq -2.45 \cdot 10^{-109}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 1.2 \cdot 10^{-186}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) ew))
        (t_2
         (fabs
          (fma
           (* (- eh) (sin t))
           (sin (atan (/ (* (- eh) t) ew)))
           (* (cos (atan (* (/ t ew) eh))) t_1)))))
   (if (<= eh -2.45e-109)
     t_2
     (if (<= eh 1.2e-186)
       (fabs (* (cos (atan (* (/ (sin t) ew) (/ eh (cos t))))) t_1))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * ew;
	double t_2 = fabs(fma((-eh * sin(t)), sin(atan(((-eh * t) / ew))), (cos(atan(((t / ew) * eh))) * t_1)));
	double tmp;
	if (eh <= -2.45e-109) {
		tmp = t_2;
	} else if (eh <= 1.2e-186) {
		tmp = fabs((cos(atan(((sin(t) / ew) * (eh / cos(t))))) * t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * ew)
	t_2 = abs(fma(Float64(Float64(-eh) * sin(t)), sin(atan(Float64(Float64(Float64(-eh) * t) / ew))), Float64(cos(atan(Float64(Float64(t / ew) * eh))) * t_1)))
	tmp = 0.0
	if (eh <= -2.45e-109)
		tmp = t_2;
	elseif (eh <= 1.2e-186)
		tmp = abs(Float64(cos(atan(Float64(Float64(sin(t) / ew) * Float64(eh / cos(t))))) * t_1));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[ArcTan[N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.45e-109], t$95$2, If[LessEqual[eh, 1.2e-186], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.44999999999999999e-109 or 1.20000000000000002e-186 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 99.0

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

   5. Applied rewrites 99.0%

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 93.3

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

   10. Applied rewrites 93.3%

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

   if -2.44999999999999999e-109 < eh < 1.20000000000000002e-186

   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

   4. Applied rewrites 59.8%

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

   5. Taylor expanded in ew around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 99.9

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

   7. Applied rewrites 99.9%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.45 \cdot 10^{-109}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left(-eh\right) \cdot \sin t, \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right), \cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right) \cdot \left(\cos t \cdot ew\right)\right)\right|\\ \mathbf{elif}\;eh \leq 1.2 \cdot 10^{-186}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left(-eh\right) \cdot \sin t, \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right), \cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right) \cdot \left(\cos t \cdot ew\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\cos \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \mathbf{if}\;ew \leq -1.85 \cdot 10^{-177}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 7 \cdot 10^{-70}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (* (cos (atan (* (/ (sin t) ew) (/ eh (cos t))))) (* (cos t) ew)))))
   (if (<= ew -1.85e-177)
     t_1
     (if (<= ew 7e-70)
       (fabs (* (* (sin t) eh) (sin (atan (/ (* eh t) ew)))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((cos(atan(((sin(t) / ew) * (eh / cos(t))))) * (cos(t) * ew)));
	double tmp;
	if (ew <= -1.85e-177) {
		tmp = t_1;
	} else if (ew <= 7e-70) {
		tmp = fabs(((sin(t) * eh) * sin(atan(((eh * 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((cos(atan(((sin(t) / ew) * (eh / cos(t))))) * (cos(t) * ew)))
    if (ew <= (-1.85d-177)) then
        tmp = t_1
    else if (ew <= 7d-70) then
        tmp = abs(((sin(t) * eh) * sin(atan(((eh * 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((Math.cos(Math.atan(((Math.sin(t) / ew) * (eh / Math.cos(t))))) * (Math.cos(t) * ew)));
	double tmp;
	if (ew <= -1.85e-177) {
		tmp = t_1;
	} else if (ew <= 7e-70) {
		tmp = Math.abs(((Math.sin(t) * eh) * Math.sin(Math.atan(((eh * t) / ew)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.cos(math.atan(((math.sin(t) / ew) * (eh / math.cos(t))))) * (math.cos(t) * ew)))
	tmp = 0
	if ew <= -1.85e-177:
		tmp = t_1
	elif ew <= 7e-70:
		tmp = math.fabs(((math.sin(t) * eh) * math.sin(math.atan(((eh * t) / ew)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(cos(atan(Float64(Float64(sin(t) / ew) * Float64(eh / cos(t))))) * Float64(cos(t) * ew)))
	tmp = 0.0
	if (ew <= -1.85e-177)
		tmp = t_1;
	elseif (ew <= 7e-70)
		tmp = abs(Float64(Float64(sin(t) * eh) * sin(atan(Float64(Float64(eh * t) / ew)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((cos(atan(((sin(t) / ew) * (eh / cos(t))))) * (cos(t) * ew)));
	tmp = 0.0;
	if (ew <= -1.85e-177)
		tmp = t_1;
	elseif (ew <= 7e-70)
		tmp = abs(((sin(t) * eh) * sin(atan(((eh * t) / ew)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.85e-177], t$95$1, If[LessEqual[ew, 7e-70], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.84999999999999997e-177 or 6.99999999999999949e-70 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in ew around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 77.1

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

   7. Applied rewrites 77.1%

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

   if -1.84999999999999997e-177 < ew < 6.99999999999999949e-70

   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 rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. cos-mult N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in ew around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 76.8%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 76.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.85 \cdot 10^{-177}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \mathbf{elif}\;ew \leq 7 \cdot 10^{-70}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right)\right|\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1800:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (* (sin t) eh) (sin (atan (/ (* eh t) ew)))))))
   (if (<= t -3.5e-72) t_1 (if (<= t 1800.0) (fabs (/ ew 1.0)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((sin(t) * eh) * sin(atan(((eh * t) / ew)))));
	double tmp;
	if (t <= -3.5e-72) {
		tmp = t_1;
	} else if (t <= 1800.0) {
		tmp = fabs((ew / 1.0));
	} 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(((sin(t) * eh) * sin(atan(((eh * t) / ew)))))
    if (t <= (-3.5d-72)) then
        tmp = t_1
    else if (t <= 1800.0d0) then
        tmp = abs((ew / 1.0d0))
    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(((Math.sin(t) * eh) * Math.sin(Math.atan(((eh * t) / ew)))));
	double tmp;
	if (t <= -3.5e-72) {
		tmp = t_1;
	} else if (t <= 1800.0) {
		tmp = Math.abs((ew / 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs(((math.sin(t) * eh) * math.sin(math.atan(((eh * t) / ew)))))
	tmp = 0
	if t <= -3.5e-72:
		tmp = t_1
	elif t <= 1800.0:
		tmp = math.fabs((ew / 1.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(sin(t) * eh) * sin(atan(Float64(Float64(eh * t) / ew)))))
	tmp = 0.0
	if (t <= -3.5e-72)
		tmp = t_1;
	elseif (t <= 1800.0)
		tmp = abs(Float64(ew / 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs(((sin(t) * eh) * sin(atan(((eh * t) / ew)))));
	tmp = 0.0;
	if (t <= -3.5e-72)
		tmp = t_1;
	elseif (t <= 1800.0)
		tmp = abs((ew / 1.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -3.5e-72], t$95$1, If[LessEqual[t, 1800.0], N[Abs[N[(ew / 1.0), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.5e-72 or 1800 < 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 rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. cos-mult N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

   6. Applied rewrites 98.4%

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

   7. Taylor expanded in ew around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   9. Applied rewrites 51.2%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 51.5%

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

   if -3.5e-72 < t < 1800

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 78.6%

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

   10. Applied rewrites 78.5%

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

   11. Taylor expanded in ew around inf

       \[\leadsto \left|\frac{ew}{1}\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 78.7%

       \[\leadsto \left|\frac{ew}{1}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-72}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right)\right|\\ \mathbf{elif}\;t \leq 1800:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.2% accurate, 61.6× speedup

Math

\[\begin{array}{l} \\ \left|\frac{ew}{1}\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (/ ew 1.0)))

C

double code(double eh, double ew, double t) {
	return fabs((ew / 1.0));
}

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 / 1.0d0))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((ew / 1.0));
}

Python

def code(eh, ew, t):
	return math.fabs((ew / 1.0))

Julia

function code(eh, ew, t)
	return abs(Float64(ew / 1.0))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((ew / 1.0));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew / 1.0), $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. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 44.7%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 43.6%

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

10. Applied rewrites 42.7%

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

11. Taylor expanded in ew around inf

    \[\leadsto \left|\frac{ew}{1}\right| \]
12. Step-by-step derivation

13. Applied rewrites 44.8%

    \[\leadsto \left|\frac{ew}{1}\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024268 
(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)))))))