Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 38.0s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 87.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ \mathbf{if}\;eh \leq -2.4 \cdot 10^{+59} \lor \neg \left(eh \leq 3.6 \cdot 10^{+112}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot t\_1, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} t\_1}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)))
   (if (or (<= eh -2.4e+59) (not (<= eh 3.6e+112)))
     (fabs
      (*
       (sin (atan (* (/ (fma -0.5 (* t t) 1.0) ew) (/ eh (sin t)))))
       (* (cos t) eh)))
     (fabs (/ (fma (* (cos t) t_1) eh (* (sin t) ew)) (cosh (asinh t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double tmp;
	if ((eh <= -2.4e+59) || !(eh <= 3.6e+112)) {
		tmp = fabs((sin(atan(((fma(-0.5, (t * t), 1.0) / ew) * (eh / sin(t))))) * (cos(t) * eh)));
	} else {
		tmp = fabs((fma((cos(t) * t_1), eh, (sin(t) * ew)) / cosh(asinh(t_1))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	tmp = 0.0
	if ((eh <= -2.4e+59) || !(eh <= 3.6e+112))
		tmp = abs(Float64(sin(atan(Float64(Float64(fma(-0.5, Float64(t * t), 1.0) / ew) * Float64(eh / sin(t))))) * Float64(cos(t) * eh)));
	else
		tmp = abs(Float64(fma(Float64(cos(t) * t_1), eh, Float64(sin(t) * ew)) / cosh(asinh(t_1))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -2.4e+59], N[Not[LessEqual[eh, 3.6e+112]], $MachinePrecision]], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(-0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * t$95$1), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.4000000000000002e59 or 3.6e112 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 92.5%

      \[\leadsto \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right| \]

   if -2.4000000000000002e59 < eh < 3.6e112

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 87.9%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.4 \cdot 10^{+59} \lor \neg \left(eh \leq 3.6 \cdot 10^{+112}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ \mathbf{if}\;eh \leq -2.5 \cdot 10^{-21} \lor \neg \left(eh \leq 1.05 \cdot 10^{-20}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot t\_1, eh, \sin t \cdot ew\right)}{\sqrt{1 + {t\_1}^{2}}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)))
   (if (or (<= eh -2.5e-21) (not (<= eh 1.05e-20)))
     (fabs
      (*
       (sin (atan (* (/ (fma -0.5 (* t t) 1.0) ew) (/ eh (sin t)))))
       (* (cos t) eh)))
     (fabs
      (/
       (fma (* (cos t) t_1) eh (* (sin t) ew))
       (sqrt (+ 1.0 (pow t_1 2.0))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double tmp;
	if ((eh <= -2.5e-21) || !(eh <= 1.05e-20)) {
		tmp = fabs((sin(atan(((fma(-0.5, (t * t), 1.0) / ew) * (eh / sin(t))))) * (cos(t) * eh)));
	} else {
		tmp = fabs((fma((cos(t) * t_1), eh, (sin(t) * ew)) / sqrt((1.0 + pow(t_1, 2.0)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	tmp = 0.0
	if ((eh <= -2.5e-21) || !(eh <= 1.05e-20))
		tmp = abs(Float64(sin(atan(Float64(Float64(fma(-0.5, Float64(t * t), 1.0) / ew) * Float64(eh / sin(t))))) * Float64(cos(t) * eh)));
	else
		tmp = abs(Float64(fma(Float64(cos(t) * t_1), eh, Float64(sin(t) * ew)) / sqrt(Float64(1.0 + (t_1 ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -2.5e-21], N[Not[LessEqual[eh, 1.05e-20]], $MachinePrecision]], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(-0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * t$95$1), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.49999999999999986e-21 or 1.0499999999999999e-20 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 84.3

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

   5. Applied rewrites 84.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 84.5%

      \[\leadsto \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right| \]

   if -2.49999999999999986e-21 < eh < 1.0499999999999999e-20

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 93.8%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 92.8

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

   6. Applied rewrites 92.8%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.5 \cdot 10^{-21} \lor \neg \left(eh \leq 1.05 \cdot 10^{-20}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2}}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -3.4 \cdot 10^{-17} \lor \neg \left(eh \leq 1.05 \cdot 10^{-20}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew}}{t}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -3.4e-17) (not (<= eh 1.05e-20)))
   (fabs
    (*
     (sin (atan (* (/ (fma -0.5 (* t t) 1.0) ew) (/ eh (sin t)))))
     (* (cos t) eh)))
   (fabs
    (/
     (fma (/ (/ eh ew) t) eh (* (sin t) ew))
     (cosh (asinh (/ (/ eh (tan t)) ew)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3.4e-17) || !(eh <= 1.05e-20)) {
		tmp = fabs((sin(atan(((fma(-0.5, (t * t), 1.0) / ew) * (eh / sin(t))))) * (cos(t) * eh)));
	} else {
		tmp = fabs((fma(((eh / ew) / t), eh, (sin(t) * ew)) / cosh(asinh(((eh / tan(t)) / ew)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -3.4e-17) || !(eh <= 1.05e-20))
		tmp = abs(Float64(sin(atan(Float64(Float64(fma(-0.5, Float64(t * t), 1.0) / ew) * Float64(eh / sin(t))))) * Float64(cos(t) * eh)));
	else
		tmp = abs(Float64(fma(Float64(Float64(eh / ew) / t), eh, Float64(sin(t) * ew)) / cosh(asinh(Float64(Float64(eh / tan(t)) / ew)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -3.4e-17], N[Not[LessEqual[eh, 1.05e-20]], $MachinePrecision]], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(-0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.3999999999999998e-17 or 1.0499999999999999e-20 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 84.3

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

   5. Applied rewrites 84.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 84.5%

      \[\leadsto \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right| \]

   if -3.3999999999999998e-17 < eh < 1.0499999999999999e-20

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 84.0

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

   7. Applied rewrites 84.0%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.4 \cdot 10^{-17} \lor \neg \left(eh \leq 1.05 \cdot 10^{-20}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew}}{t}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -7.7 \cdot 10^{-22} \lor \neg \left(eh \leq 8.8 \cdot 10^{-21}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew}}{t}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2}}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -7.7e-22) (not (<= eh 8.8e-21)))
   (fabs
    (*
     (sin (atan (* (/ (fma -0.5 (* t t) 1.0) ew) (/ eh (sin t)))))
     (* (cos t) eh)))
   (fabs
    (/
     (fma (/ (/ eh ew) t) eh (* (sin t) ew))
     (sqrt (+ 1.0 (pow (/ (/ eh (tan t)) ew) 2.0)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -7.7e-22) || !(eh <= 8.8e-21)) {
		tmp = fabs((sin(atan(((fma(-0.5, (t * t), 1.0) / ew) * (eh / sin(t))))) * (cos(t) * eh)));
	} else {
		tmp = fabs((fma(((eh / ew) / t), eh, (sin(t) * ew)) / sqrt((1.0 + pow(((eh / tan(t)) / ew), 2.0)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -7.7e-22) || !(eh <= 8.8e-21))
		tmp = abs(Float64(sin(atan(Float64(Float64(fma(-0.5, Float64(t * t), 1.0) / ew) * Float64(eh / sin(t))))) * Float64(cos(t) * eh)));
	else
		tmp = abs(Float64(fma(Float64(Float64(eh / ew) / t), eh, Float64(sin(t) * ew)) / sqrt(Float64(1.0 + (Float64(Float64(eh / tan(t)) / ew) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -7.7e-22], N[Not[LessEqual[eh, 8.8e-21]], $MachinePrecision]], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(-0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -7.7000000000000002e-22 or 8.8000000000000002e-21 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 84.3

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

   5. Applied rewrites 84.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 84.5%

      \[\leadsto \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right| \]

   if -7.7000000000000002e-22 < eh < 8.8000000000000002e-21

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 84.0

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

   7. Applied rewrites 84.0%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-prod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lift-cosh.f64 N/A

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

      5. lift-asinh.f64 N/A

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

      6. cosh-asinh N/A

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

      7. +-commutative N/A

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

      8. lift-cosh.f64 N/A

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

      9. lift-asinh.f64 N/A

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

      10. cosh-asinh N/A

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

      11. +-commutative N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-+.f64 N/A

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

   9. Applied rewrites 82.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7.7 \cdot 10^{-22} \lor \neg \left(eh \leq 8.8 \cdot 10^{-21}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew}}{t}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2}}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ \mathbf{if}\;eh \leq -7.7 \cdot 10^{-22} \lor \neg \left(eh \leq 8.8 \cdot 10^{-21}\right):\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} t\_1 \cdot \cos t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew}}{t}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {t\_1}^{2}}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)))
   (if (or (<= eh -7.7e-22) (not (<= eh 8.8e-21)))
     (fabs (* (* (tanh (asinh t_1)) (cos t)) eh))
     (fabs
      (/
       (fma (/ (/ eh ew) t) eh (* (sin t) ew))
       (sqrt (+ 1.0 (pow t_1 2.0))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double tmp;
	if ((eh <= -7.7e-22) || !(eh <= 8.8e-21)) {
		tmp = fabs(((tanh(asinh(t_1)) * cos(t)) * eh));
	} else {
		tmp = fabs((fma(((eh / ew) / t), eh, (sin(t) * ew)) / sqrt((1.0 + pow(t_1, 2.0)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	tmp = 0.0
	if ((eh <= -7.7e-22) || !(eh <= 8.8e-21))
		tmp = abs(Float64(Float64(tanh(asinh(t_1)) * cos(t)) * eh));
	else
		tmp = abs(Float64(fma(Float64(Float64(eh / ew) / t), eh, Float64(sin(t) * ew)) / sqrt(Float64(1.0 + (t_1 ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -7.7e-22], N[Not[LessEqual[eh, 8.8e-21]], $MachinePrecision]], N[Abs[N[(N[(N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -7.7000000000000002e-22 or 8.8000000000000002e-21 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 84.3

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

   5. Applied rewrites 84.3%

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

   7. Applied rewrites 84.3%

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

   if -7.7000000000000002e-22 < eh < 8.8000000000000002e-21

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 84.0

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

   7. Applied rewrites 84.0%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-prod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lift-cosh.f64 N/A

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

      5. lift-asinh.f64 N/A

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

      6. cosh-asinh N/A

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

      7. +-commutative N/A

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

      8. lift-cosh.f64 N/A

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

      9. lift-asinh.f64 N/A

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

      10. cosh-asinh N/A

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

      11. +-commutative N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-+.f64 N/A

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

   9. Applied rewrites 82.9%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7.7 \cdot 10^{-22} \lor \neg \left(eh \leq 8.8 \cdot 10^{-21}\right):\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot \cos t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew}}{t}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2}}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.5 \cdot 10^{-17} \lor \neg \left(eh \leq 1.2 \cdot 10^{-19}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew}}{t}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2}}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -1.5e-17) (not (<= eh 1.2e-19)))
   (fabs (* (sin (atan (* (/ (fma -0.5 (* t t) 1.0) ew) (/ eh (sin t))))) eh))
   (fabs
    (/
     (fma (/ (/ eh ew) t) eh (* (sin t) ew))
     (sqrt (+ 1.0 (pow (/ (/ eh (tan t)) ew) 2.0)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.5e-17) || !(eh <= 1.2e-19)) {
		tmp = fabs((sin(atan(((fma(-0.5, (t * t), 1.0) / ew) * (eh / sin(t))))) * eh));
	} else {
		tmp = fabs((fma(((eh / ew) / t), eh, (sin(t) * ew)) / sqrt((1.0 + pow(((eh / tan(t)) / ew), 2.0)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -1.5e-17) || !(eh <= 1.2e-19))
		tmp = abs(Float64(sin(atan(Float64(Float64(fma(-0.5, Float64(t * t), 1.0) / ew) * Float64(eh / sin(t))))) * eh));
	else
		tmp = abs(Float64(fma(Float64(Float64(eh / ew) / t), eh, Float64(sin(t) * ew)) / sqrt(Float64(1.0 + (Float64(Float64(eh / tan(t)) / ew) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -1.5e-17], N[Not[LessEqual[eh, 1.2e-19]], $MachinePrecision]], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(-0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.50000000000000003e-17 or 1.20000000000000011e-19 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 59.7

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

   5. Applied rewrites 59.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 59.9%

      \[\leadsto \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]

   if -1.50000000000000003e-17 < eh < 1.20000000000000011e-19

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 84.0

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

   7. Applied rewrites 84.0%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-prod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lift-cosh.f64 N/A

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

      5. lift-asinh.f64 N/A

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

      6. cosh-asinh N/A

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

      7. +-commutative N/A

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

      8. lift-cosh.f64 N/A

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

      9. lift-asinh.f64 N/A

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

      10. cosh-asinh N/A

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

      11. +-commutative N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-+.f64 N/A

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

   9. Applied rewrites 82.9%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.5 \cdot 10^{-17} \lor \neg \left(eh \leq 1.2 \cdot 10^{-19}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew}}{t}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2}}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -2 \cdot 10^{-21} \lor \neg \left(eh \leq 1.05 \cdot 10^{-20}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -2e-21) (not (<= eh 1.05e-20)))
   (fabs (* (sin (atan (* (/ (fma -0.5 (* t t) 1.0) ew) (/ eh (sin t))))) eh))
   (fabs (* ew (sin t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -2e-21) || !(eh <= 1.05e-20)) {
		tmp = fabs((sin(atan(((fma(-0.5, (t * t), 1.0) / ew) * (eh / sin(t))))) * eh));
	} else {
		tmp = fabs((ew * sin(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -2e-21) || !(eh <= 1.05e-20))
		tmp = abs(Float64(sin(atan(Float64(Float64(fma(-0.5, Float64(t * t), 1.0) / ew) * Float64(eh / sin(t))))) * eh));
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -2e-21], N[Not[LessEqual[eh, 1.05e-20]], $MachinePrecision]], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(-0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.99999999999999982e-21 or 1.0499999999999999e-20 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 59.7

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

   5. Applied rewrites 59.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 59.9%

      \[\leadsto \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]

   if -1.99999999999999982e-21 < eh < 1.0499999999999999e-20

   1. Initial program 99.8%

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

   4. Applied rewrites 55.2%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 69.7

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

   7. Applied rewrites 69.7%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2 \cdot 10^{-21} \lor \neg \left(eh \leq 1.05 \cdot 10^{-20}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{-62} \lor \neg \left(t \leq 0.00365\right):\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew}}{t}, eh, \sin t \cdot ew\right)}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -5.4e-62) (not (<= t 0.00365)))
   (fabs (/ (fma (/ (/ eh ew) t) eh (* (sin t) ew)) 1.0))
   (fabs (* (tanh (asinh (/ (/ eh (tan t)) ew))) eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -5.4e-62) || !(t <= 0.00365)) {
		tmp = fabs((fma(((eh / ew) / t), eh, (sin(t) * ew)) / 1.0));
	} else {
		tmp = fabs((tanh(asinh(((eh / tan(t)) / ew))) * eh));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -5.4e-62) || !(t <= 0.00365))
		tmp = abs(Float64(fma(Float64(Float64(eh / ew) / t), eh, Float64(sin(t) * ew)) / 1.0));
	else
		tmp = abs(Float64(tanh(asinh(Float64(Float64(eh / tan(t)) / ew))) * eh));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -5.4e-62], N[Not[LessEqual[t, 0.00365]], $MachinePrecision]], N[Abs[N[(N[(N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Tanh[N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.40000000000000039e-62 or 0.00365000000000000003 < t

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 74.4%

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

   5. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 55.0

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

   7. Applied rewrites 55.0%

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

   8. Taylor expanded in eh around 0

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

   10. Applied rewrites 54.0%

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

   if -5.40000000000000039e-62 < t < 0.00365000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 74.7

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

   5. Applied rewrites 74.7%

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

   7. Applied rewrites 74.7%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{-62} \lor \neg \left(t \leq 0.00365\right):\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew}}{t}, eh, \sin t \cdot ew\right)}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -3.8 \cdot 10^{+66}:\\ \;\;\;\;\tanh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot eh\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew}}{t}, eh, \sin t \cdot ew\right)}{1}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -3.8e+66)
   (* (tanh (asinh (/ (/ eh (tan t)) ew))) eh)
   (fabs (/ (fma (/ (/ eh ew) t) eh (* (sin t) ew)) 1.0))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -3.8e+66) {
		tmp = tanh(asinh(((eh / tan(t)) / ew))) * eh;
	} else {
		tmp = fabs((fma(((eh / ew) / t), eh, (sin(t) * ew)) / 1.0));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -3.8e+66)
		tmp = Float64(tanh(asinh(Float64(Float64(eh / tan(t)) / ew))) * eh);
	else
		tmp = abs(Float64(fma(Float64(Float64(eh / ew) / t), eh, Float64(sin(t) * ew)) / 1.0));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -3.8e+66], N[(N[Tanh[N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision], N[Abs[N[(N[(N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.8000000000000002e66

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 70.0

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

   5. Applied rewrites 70.0%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 45.8

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

   7. Applied rewrites 45.8%

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

   if -3.8000000000000002e66 < eh

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 72.6%

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

   5. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 61.7

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

   7. Applied rewrites 61.7%

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

   8. Taylor expanded in eh around 0

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

   10. Applied rewrites 47.8%

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

Alternative 11: 43.4% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-atan.f64 N/A

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

   6. sin-atan N/A

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

   7. associate-*r/ N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

4. Applied rewrites 64.2%

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

5. Taylor expanded in t around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 54.1

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

7. Applied rewrites 54.1%

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

8. Taylor expanded in eh around 0

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

10. Applied rewrites 41.8%

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

Alternative 12: 42.4% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(Float64(ew * sin(t)))
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 36.7%

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

5. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 40.8

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

7. Applied rewrites 40.8%

   \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
8. Add Preprocessing

Alternative 13: 19.1% accurate, 108.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(t * ew), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 36.7%

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

5. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 40.8

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

7. Applied rewrites 40.8%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 19.2%

    \[\leadsto \left|t \cdot \color{blue}{ew}\right| \]
11. Add Preprocessing

Alternative 14: 10.8% accurate, 145.0× speedup

Math

\[\begin{array}{l} \\ t \cdot ew \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double eh, double ew, double t) {
	return t * ew;
}

Python

def code(eh, ew, t):
	return t * ew

Julia

function code(eh, ew, t)
	return Float64(t * ew)
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[(t * ew), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 36.7%

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

5. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 40.8

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

7. Applied rewrites 40.8%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 19.2%

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

    1. lift-fabs.f64 N/A

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

    2. rem-sqrt-square-rev N/A

       \[\leadsto \color{blue}{\sqrt{\left(t \cdot ew\right) \cdot \left(t \cdot ew\right)}} \]

    3. sqrt-prod N/A

       \[\leadsto \color{blue}{\sqrt{t \cdot ew} \cdot \sqrt{t \cdot ew}} \]

    4. rem-square-sqrt 12.3

       \[\leadsto \color{blue}{t \cdot ew} \]

12. Applied rewrites 12.3%

    \[\leadsto \color{blue}{t \cdot ew} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))