Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 13.8s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(eh * Float64(tan(t) / ew)))
	return abs(fma(Float64(sin(t_1) * eh), sin(t), Float64(Float64(cos(t) * ew) * cos(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[Sin[t$95$1], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[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%

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

   1. lift-fma.f64 N/A

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

6. Applied rewrites 99.8%

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

Alternative 2: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lift-fma.f64 N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in t around 0

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

   1. lower-/.f64 98.9

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

9. Applied rewrites 98.9%

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

Alternative 3: 74.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \frac{\tan t}{ew}\\ \mathbf{if}\;eh \leq -4.3 \cdot 10^{-17} \lor \neg \left(eh \leq 2.9 \cdot 10^{+94}\right):\\ \;\;\;\;\left|\left(\sin t \cdot \left(-eh\right)\right) \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\sin t}{\cos t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-ew, \cos t, \left(t\_1 \cdot eh\right) \cdot \left(-\sin t\right)\right) \cdot \cos \tan^{-1} t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (/ (tan t) ew))))
   (if (or (<= eh -4.3e-17) (not (<= eh 2.9e+94)))
     (fabs
      (*
       (* (sin t) (- eh))
       (sin (atan (* (- eh) (/ (sin t) (* (cos t) ew)))))))
     (fabs
      (* (fma (- ew) (cos t) (* (* t_1 eh) (- (sin t)))) (cos (atan t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * (tan(t) / ew);
	double tmp;
	if ((eh <= -4.3e-17) || !(eh <= 2.9e+94)) {
		tmp = fabs(((sin(t) * -eh) * sin(atan((-eh * (sin(t) / (cos(t) * ew)))))));
	} else {
		tmp = fabs((fma(-ew, cos(t), ((t_1 * eh) * -sin(t))) * cos(atan(t_1))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * Float64(tan(t) / ew))
	tmp = 0.0
	if ((eh <= -4.3e-17) || !(eh <= 2.9e+94))
		tmp = abs(Float64(Float64(sin(t) * Float64(-eh)) * sin(atan(Float64(Float64(-eh) * Float64(sin(t) / Float64(cos(t) * ew)))))));
	else
		tmp = abs(Float64(fma(Float64(-ew), cos(t), Float64(Float64(t_1 * eh) * Float64(-sin(t)))) * cos(atan(t_1))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -4.3e-17], N[Not[LessEqual[eh, 2.9e+94]], $MachinePrecision]], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * (-eh)), $MachinePrecision] * N[Sin[N[ArcTan[N[((-eh) * N[(N[Sin[t], $MachinePrecision] / N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision] + N[(N[(t$95$1 * eh), $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.30000000000000023e-17 or 2.8999999999999998e94 < 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

      \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)}\right| \]
   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 20.3%

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

   6. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-atan.f64 N/A

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

      13. mul-1-neg N/A

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

      14. associate-/l* N/A

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

      15. distribute-lft-neg-in N/A

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

   8. Applied rewrites 75.5%

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

   if -4.30000000000000023e-17 < eh < 2.8999999999999998e94

   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 84.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. /-rgt-identity N/A

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

      5. lower-*.f64 84.3

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

   6. Applied rewrites 84.3%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.3 \cdot 10^{-17} \lor \neg \left(eh \leq 2.9 \cdot 10^{+94}\right):\\ \;\;\;\;\left|\left(\sin t \cdot \left(-eh\right)\right) \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\sin t}{\cos t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(-ew, \cos t, \left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \left(-\sin t\right)\right) \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot ew\\ t_2 := \tan^{-1} \left(\left(-eh\right) \cdot \frac{\sin t}{t\_1}\right)\\ \mathbf{if}\;eh \leq -4.3 \cdot 10^{-17} \lor \neg \left(eh \leq 2.9 \cdot 10^{+94}\right):\\ \;\;\;\;\left|\left(\sin t \cdot \left(-eh\right)\right) \cdot \sin t\_2\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1 \cdot \cos t\_2\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) ew)) (t_2 (atan (* (- eh) (/ (sin t) t_1)))))
   (if (or (<= eh -4.3e-17) (not (<= eh 2.9e+94)))
     (fabs (* (* (sin t) (- eh)) (sin t_2)))
     (fabs (* t_1 (cos t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * ew;
	double t_2 = atan((-eh * (sin(t) / t_1)));
	double tmp;
	if ((eh <= -4.3e-17) || !(eh <= 2.9e+94)) {
		tmp = fabs(((sin(t) * -eh) * sin(t_2)));
	} else {
		tmp = fabs((t_1 * cos(t_2)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = cos(t) * ew
    t_2 = atan((-eh * (sin(t) / t_1)))
    if ((eh <= (-4.3d-17)) .or. (.not. (eh <= 2.9d+94))) then
        tmp = abs(((sin(t) * -eh) * sin(t_2)))
    else
        tmp = abs((t_1 * cos(t_2)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.cos(t) * ew;
	double t_2 = Math.atan((-eh * (Math.sin(t) / t_1)));
	double tmp;
	if ((eh <= -4.3e-17) || !(eh <= 2.9e+94)) {
		tmp = Math.abs(((Math.sin(t) * -eh) * Math.sin(t_2)));
	} else {
		tmp = Math.abs((t_1 * Math.cos(t_2)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.cos(t) * ew
	t_2 = math.atan((-eh * (math.sin(t) / t_1)))
	tmp = 0
	if (eh <= -4.3e-17) or not (eh <= 2.9e+94):
		tmp = math.fabs(((math.sin(t) * -eh) * math.sin(t_2)))
	else:
		tmp = math.fabs((t_1 * math.cos(t_2)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * ew)
	t_2 = atan(Float64(Float64(-eh) * Float64(sin(t) / t_1)))
	tmp = 0.0
	if ((eh <= -4.3e-17) || !(eh <= 2.9e+94))
		tmp = abs(Float64(Float64(sin(t) * Float64(-eh)) * sin(t_2)));
	else
		tmp = abs(Float64(t_1 * cos(t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = cos(t) * ew;
	t_2 = atan((-eh * (sin(t) / t_1)));
	tmp = 0.0;
	if ((eh <= -4.3e-17) || ~((eh <= 2.9e+94)))
		tmp = abs(((sin(t) * -eh) * sin(t_2)));
	else
		tmp = abs((t_1 * cos(t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[((-eh) * N[(N[Sin[t], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[eh, -4.3e-17], N[Not[LessEqual[eh, 2.9e+94]], $MachinePrecision]], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * (-eh)), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.30000000000000023e-17 or 2.8999999999999998e94 < 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

      \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)}\right| \]
   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 20.3%

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

   6. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-atan.f64 N/A

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

      13. mul-1-neg N/A

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

      14. associate-/l* N/A

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

      15. distribute-lft-neg-in N/A

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

   8. Applied rewrites 75.5%

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

   if -4.30000000000000023e-17 < eh < 2.8999999999999998e94

   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 53.1%

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

   6. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. mul-1-neg N/A

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

      9. associate-/l* N/A

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

      10. distribute-lft-neg-in N/A

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

      11. neg-mul-1 N/A

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

      12. lower-*.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-sin.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 83.3

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

   8. Applied rewrites 83.3%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.3 \cdot 10^{-17} \lor \neg \left(eh \leq 2.9 \cdot 10^{+94}\right):\\ \;\;\;\;\left|\left(\sin t \cdot \left(-eh\right)\right) \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\sin t}{\cos t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\sin t}{\cos t \cdot ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs(((-ew * cos(t)) / pow(cos(atan(((tan(t) / ew) * eh))), -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 * cos(t)) / (cos(atan(((tan(t) / ew) * eh))) ** (-1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], -1.0], $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 60.4%

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

5. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-cos.f64 57.8

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

7. Applied rewrites 57.8%

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

8. Final simplification 57.8%

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

Alternative 6: 62.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. 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 38.1%

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

6. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-atan.f64 N/A

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

   8. mul-1-neg N/A

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

   9. associate-/l* N/A

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

   10. distribute-lft-neg-in N/A

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

   11. neg-mul-1 N/A

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

   12. lower-*.f64 N/A

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

   13. neg-mul-1 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-sin.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 57.8

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

8. Applied rewrites 57.8%

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

Alternative 7: 44.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-15}:\\ \;\;\;\;\left|\frac{\cos t \cdot ew - \mathsf{fma}\left(\mathsf{fma}\left(\left(eh \cdot eh\right) \cdot \frac{t \cdot t}{ew}, -0.08611111111111111, \frac{eh \cdot eh}{ew} \cdot -0.16666666666666666\right), t \cdot t, \frac{\left(-eh\right) \cdot eh}{ew}\right) \cdot \left(t \cdot t\right)}{\frac{-1}{\cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t -2.9e-15)
   (fabs
    (/
     (-
      (* (cos t) ew)
      (*
       (fma
        (fma
         (* (* eh eh) (/ (* t t) ew))
         -0.08611111111111111
         (* (/ (* eh eh) ew) -0.16666666666666666))
        (* t t)
        (/ (* (- eh) eh) ew))
       (* t t)))
     (/ -1.0 (cos (atan (* (/ t ew) eh))))))
   (fabs (/ ew 1.0))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -2.9e-15) {
		tmp = fabs((((cos(t) * ew) - (fma(fma(((eh * eh) * ((t * t) / ew)), -0.08611111111111111, (((eh * eh) / ew) * -0.16666666666666666)), (t * t), ((-eh * eh) / ew)) * (t * t))) / (-1.0 / cos(atan(((t / ew) * eh))))));
	} else {
		tmp = fabs((ew / 1.0));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= -2.9e-15)
		tmp = abs(Float64(Float64(Float64(cos(t) * ew) - Float64(fma(fma(Float64(Float64(eh * eh) * Float64(Float64(t * t) / ew)), -0.08611111111111111, Float64(Float64(Float64(eh * eh) / ew) * -0.16666666666666666)), Float64(t * t), Float64(Float64(Float64(-eh) * eh) / ew)) * Float64(t * t))) / Float64(-1.0 / cos(atan(Float64(Float64(t / ew) * eh))))));
	else
		tmp = abs(Float64(ew / 1.0));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, -2.9e-15], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] - N[(N[(N[(N[(N[(eh * eh), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * -0.08611111111111111 + N[(N[(N[(eh * eh), $MachinePrecision] / ew), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(N[((-eh) * eh), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / N[Cos[N[ArcTan[N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew / 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.90000000000000019e-15

   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 48.8%

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

   5. Taylor expanded in t around 0

      \[\leadsto \left|\frac{\color{blue}{{t}^{2} \cdot \left(-1 \cdot \frac{{eh}^{2}}{ew} + {t}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{{eh}^{2}}{ew} + \frac{-31}{360} \cdot \frac{{eh}^{2} \cdot {t}^{2}}{ew}\right)\right)} - \cos t \cdot ew}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right| \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left|\frac{\color{blue}{\left(-1 \cdot \frac{{eh}^{2}}{ew} + {t}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{{eh}^{2}}{ew} + \frac{-31}{360} \cdot \frac{{eh}^{2} \cdot {t}^{2}}{ew}\right)\right) \cdot {t}^{2}} - \cos t \cdot ew}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\frac{\color{blue}{\left(-1 \cdot \frac{{eh}^{2}}{ew} + {t}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{{eh}^{2}}{ew} + \frac{-31}{360} \cdot \frac{{eh}^{2} \cdot {t}^{2}}{ew}\right)\right) \cdot {t}^{2}} - \cos t \cdot ew}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right| \]

   7. Applied rewrites 19.9%

      \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(eh \cdot eh\right) \cdot \frac{t \cdot t}{ew}, -0.08611111111111111, \frac{eh \cdot eh}{ew} \cdot -0.16666666666666666\right), t \cdot t, \frac{eh \cdot eh}{-ew}\right) \cdot \left(t \cdot t\right)} - \cos t \cdot ew}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(eh \cdot eh\right) \cdot \frac{t \cdot t}{ew}, \frac{-31}{360}, \frac{eh \cdot eh}{ew} \cdot \frac{-1}{6}\right), t \cdot t, \frac{eh \cdot eh}{-ew}\right) \cdot \left(t \cdot t\right) - \cos t \cdot ew}{\frac{1}{\cos \tan^{-1} \left(\color{blue}{\frac{t}{ew}} \cdot eh\right)}}\right| \]
   9. Step-by-step derivation

      1. lower-/.f64 19.9

         \[\leadsto \left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(eh \cdot eh\right) \cdot \frac{t \cdot t}{ew}, -0.08611111111111111, \frac{eh \cdot eh}{ew} \cdot -0.16666666666666666\right), t \cdot t, \frac{eh \cdot eh}{-ew}\right) \cdot \left(t \cdot t\right) - \cos t \cdot ew}{\frac{1}{\cos \tan^{-1} \left(\color{blue}{\frac{t}{ew}} \cdot eh\right)}}\right| \]

   10. Applied rewrites 19.9%

       \[\leadsto \left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(\left(eh \cdot eh\right) \cdot \frac{t \cdot t}{ew}, -0.08611111111111111, \frac{eh \cdot eh}{ew} \cdot -0.16666666666666666\right), t \cdot t, \frac{eh \cdot eh}{-ew}\right) \cdot \left(t \cdot t\right) - \cos t \cdot ew}{\frac{1}{\cos \tan^{-1} \left(\color{blue}{\frac{t}{ew}} \cdot eh\right)}}\right| \]

   if -2.90000000000000019e-15 < t

   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

      \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)}\right| \]
   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 49.7%

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

   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 48.7%

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

   10. Applied rewrites 47.9%

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

   11. Taylor expanded in eh around 0

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

   13. Applied rewrites 49.8%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-15}:\\ \;\;\;\;\left|\frac{\cos t \cdot ew - \mathsf{fma}\left(\mathsf{fma}\left(\left(eh \cdot eh\right) \cdot \frac{t \cdot t}{ew}, -0.08611111111111111, \frac{eh \cdot eh}{ew} \cdot -0.16666666666666666\right), t \cdot t, \frac{\left(-eh\right) \cdot eh}{ew}\right) \cdot \left(t \cdot t\right)}{\frac{-1}{\cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.6% 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 38.1%

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

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 36.9%

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

10. Applied rewrites 35.8%

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

11. Taylor expanded in eh around 0

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

13. Applied rewrites 38.3%

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

Reproduce

herbie shell --seed 2024325 
(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)))))))