Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.6s

Alternatives: 9

Speedup: N/A×

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. 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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 99.9

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \tan t\\ \left|\mathsf{fma}\left(\frac{\sin t}{\sqrt{1 + {\left(\frac{t\_1}{eh}\right)}^{-2}}}, ew, \left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{t\_1}\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (tan t))))
   (fabs
    (fma
     (/ (sin t) (sqrt (+ 1.0 (pow (/ t_1 eh) -2.0))))
     ew
     (* (* (cos t) eh) (sin (atan (/ eh t_1))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * tan(t);
	return fabs(fma((sin(t) / sqrt((1.0 + pow((t_1 / eh), -2.0)))), ew, ((cos(t) * eh) * sin(atan((eh / t_1))))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * tan(t))
	return abs(fma(Float64(sin(t) / sqrt(Float64(1.0 + (Float64(t_1 / eh) ^ -2.0)))), ew, Float64(Float64(cos(t) * eh) * sin(atan(Float64(eh / t_1))))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[Sin[t], $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[(t$95$1 / eh), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew + N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.9%

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

   1. 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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 99.9

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-atan.f64 N/A

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

   5. cos-atan N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/l/ N/A

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

   11. lift-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. associate-/l/ N/A

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

   16. lift-*.f64 N/A

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

   17. lift-/.f64 N/A

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

8. Applied rewrites 99.9%

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

   1. lift-pow.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. inv-pow N/A

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

   5. pow-pow N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-pow.f64 N/A

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

   9. div-inv N/A

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

   10. lift-/.f64 N/A

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

   11. clear-num N/A

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

   12. associate-/l* N/A

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

   13. lift-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. metadata-eval 99.9

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

10. Applied rewrites 99.9%

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

11. Final simplification 99.9%

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

Alternative 3: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 98.7

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

5. Applied rewrites 98.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

7. Applied rewrites 98.7%

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

8. Final simplification 98.7%

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

Alternative 4: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) \cdot \left(\cos t \cdot eh\right) + \frac{\sin t \cdot ew}{\sqrt{{\left(\frac{eh}{ew \cdot t}\right)}^{2} + 1}}\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (+
   (* (sin (atan (/ (/ eh ew) (tan t)))) (* (cos t) eh))
   (/ (* (sin t) ew) (sqrt (+ (pow (/ eh (* ew t)) 2.0) 1.0))))))

C

double code(double eh, double ew, double t) {
	return fabs(((sin(atan(((eh / ew) / tan(t)))) * (cos(t) * eh)) + ((sin(t) * ew) / sqrt((pow((eh / (ew * t)), 2.0) + 1.0)))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(((sin(atan(((eh / ew) / tan(t)))) * (cos(t) * eh)) + ((sin(t) * ew) / sqrt((((eh / (ew * t)) ** 2.0d0) + 1.0d0)))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(((Math.sin(Math.atan(((eh / ew) / Math.tan(t)))) * (Math.cos(t) * eh)) + ((Math.sin(t) * ew) / Math.sqrt((Math.pow((eh / (ew * t)), 2.0) + 1.0)))));
}

Python

def code(eh, ew, t):
	return math.fabs(((math.sin(math.atan(((eh / ew) / math.tan(t)))) * (math.cos(t) * eh)) + ((math.sin(t) * ew) / math.sqrt((math.pow((eh / (ew * t)), 2.0) + 1.0)))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(sin(atan(Float64(Float64(eh / ew) / tan(t)))) * Float64(cos(t) * eh)) + Float64(Float64(sin(t) * ew) / sqrt(Float64((Float64(eh / Float64(ew * t)) ^ 2.0) + 1.0)))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((sin(atan(((eh / ew) / tan(t)))) * (cos(t) * eh)) + ((sin(t) * ew) / sqrt((((eh / (ew * t)) ^ 2.0) + 1.0)))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[Sqrt[N[(N[Power[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

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

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 98.7

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

5. Applied rewrites 98.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-atan.f64 N/A

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

   7. cos-atan N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

7. Applied rewrites 98.7%

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

8. Final simplification 98.7%

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

Alternative 5: 98.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. 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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 99.9

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-atan.f64 N/A

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

   5. cos-atan N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/l/ N/A

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

   11. lift-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. associate-/l/ N/A

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

   16. lift-*.f64 N/A

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

   17. lift-/.f64 N/A

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in eh around 0

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

    1. associate-*r* N/A

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

    2. lower-fma.f64 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. lower-cos.f64 N/A

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

    6. lower-sin.f64 N/A

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

    7. lower-atan.f64 N/A

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

    8. *-commutative N/A

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

    9. times-frac N/A

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

    10. lower-*.f64 N/A

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

    11. lower-/.f64 N/A

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

    12. lower-sin.f64 N/A

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

    13. lower-/.f64 N/A

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

    14. lower-cos.f64 N/A

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

    15. *-commutative N/A

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

    16. lower-*.f64 N/A

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

11. Applied rewrites 97.9%

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

12. Final simplification 97.9%

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

Alternative 6: 98.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. 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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 99.9

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-atan.f64 N/A

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

   5. cos-atan N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/l/ N/A

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

   11. lift-*.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. associate-/l/ N/A

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

   16. lift-*.f64 N/A

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

   17. lift-/.f64 N/A

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in ew around inf

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

    1. lower-sin.f64 97.9

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

11. Applied rewrites 97.9%

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

12. Final simplification 97.9%

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

Alternative 7: 63.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;ew \leq -2.8 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.8 \cdot 10^{+49}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\left(t \cdot t\right) \cdot ew}{eh}, ew, \left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= ew -2.8e+24)
     t_1
     (if (<= ew 1.8e+49)
       (fabs
        (fma
         (/ (* (* t t) ew) eh)
         ew
         (* (* (cos t) eh) (sin (atan (/ eh (* ew (tan t))))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (ew <= -2.8e+24) {
		tmp = t_1;
	} else if (ew <= 1.8e+49) {
		tmp = fabs(fma((((t * t) * ew) / eh), ew, ((cos(t) * eh) * sin(atan((eh / (ew * tan(t))))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (ew <= -2.8e+24)
		tmp = t_1;
	elseif (ew <= 1.8e+49)
		tmp = abs(fma(Float64(Float64(Float64(t * t) * ew) / eh), ew, Float64(Float64(cos(t) * eh) * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.8e+24], t$95$1, If[LessEqual[ew, 1.8e+49], N[Abs[N[(N[(N[(N[(t * t), $MachinePrecision] * ew), $MachinePrecision] / eh), $MachinePrecision] * ew + N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.8000000000000002e24 or 1.79999999999999998e49 < ew

   1. Initial program 99.9%

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

      1. 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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 99.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-/l/ N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in ew around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 73.7

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

   11. Applied rewrites 73.7%

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

   if -2.8000000000000002e24 < ew < 1.79999999999999998e49

   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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 99.8

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-/l/ N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 57.2

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

   11. Applied rewrites 57.2%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.8 \cdot 10^{+24}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{elif}\;ew \leq 1.8 \cdot 10^{+49}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\left(t \cdot t\right) \cdot ew}{eh}, ew, \left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;ew \leq -4.4 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 2.3 \cdot 10^{+55}:\\ \;\;\;\;\left|-eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= ew -4.4e+14) t_1 (if (<= ew 2.3e+55) (fabs (- eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (ew <= -4.4e+14) {
		tmp = t_1;
	} else if (ew <= 2.3e+55) {
		tmp = fabs(-eh);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(t) * ew))
    if (ew <= (-4.4d+14)) then
        tmp = t_1
    else if (ew <= 2.3d+55) then
        tmp = abs(-eh)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(t) * ew));
	double tmp;
	if (ew <= -4.4e+14) {
		tmp = t_1;
	} else if (ew <= 2.3e+55) {
		tmp = Math.abs(-eh);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(t) * ew))
	tmp = 0
	if ew <= -4.4e+14:
		tmp = t_1
	elif ew <= 2.3e+55:
		tmp = math.fabs(-eh)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (ew <= -4.4e+14)
		tmp = t_1;
	elseif (ew <= 2.3e+55)
		tmp = abs(Float64(-eh));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(t) * ew));
	tmp = 0.0;
	if (ew <= -4.4e+14)
		tmp = t_1;
	elseif (ew <= 2.3e+55)
		tmp = abs(-eh);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -4.4e+14], t$95$1, If[LessEqual[ew, 2.3e+55], N[Abs[(-eh)], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -4.4e14 or 2.29999999999999987e55 < ew

   1. Initial program 99.9%

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

      1. 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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 99.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-/l/ N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in ew around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 72.7

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

   11. Applied rewrites 72.7%

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

   if -4.4e14 < ew < 2.29999999999999987e55

   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. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 50.7

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

   5. Applied rewrites 50.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 47.8%

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

   10. Applied rewrites 12.4%

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

   11. Taylor expanded in eh around -inf

       \[\leadsto \left|-1 \cdot \color{blue}{eh}\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 50.9%

       \[\leadsto \left|-eh\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 42.8% accurate, 174.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[(-eh)], $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
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. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 39.2

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

5. Applied rewrites 39.2%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 36.9%

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

10. Applied rewrites 13.0%

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

11. Taylor expanded in eh around -inf

    \[\leadsto \left|-1 \cdot \color{blue}{eh}\right| \]
12. Step-by-step derivation

13. Applied rewrites 39.6%

    \[\leadsto \left|-eh\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024249 
(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))))))))