Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 21.3s

Alternatives: 11

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. 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 99.2

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

5. Applied rewrites 99.2%

   \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot t}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
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 99.2%

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

Alternative 3: 98.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. 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 99.2

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

5. Applied rewrites 99.2%

   \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot t}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
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 99.2%

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

8. Taylor expanded in ew around inf

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

10. Applied rewrites 98.2%

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

Alternative 4: 75.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{t \cdot ew}\\ t_2 := \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\\ t_3 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;t \leq -2.6 \cdot 10^{+173}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{+52}:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot t\_2\right|\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\left|\mathsf{fma}\left(t\_2 \cdot \cos t, eh, \frac{t \cdot ew}{\sqrt{\mathsf{fma}\left(t\_1, t\_1, 1\right)}}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* t ew)))
        (t_2 (sin (atan (/ eh (* (tan t) ew)))))
        (t_3 (fabs (* (sin t) ew))))
   (if (<= t -2.6e+173)
     t_3
     (if (<= t -1.4e+52)
       (fabs (* (* (cos t) eh) t_2))
       (if (<= t 1.75e-6)
         (fabs (fma (* t_2 (cos t)) eh (/ (* t ew) (sqrt (fma t_1 t_1 1.0)))))
         t_3)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (t * ew);
	double t_2 = sin(atan((eh / (tan(t) * ew))));
	double t_3 = fabs((sin(t) * ew));
	double tmp;
	if (t <= -2.6e+173) {
		tmp = t_3;
	} else if (t <= -1.4e+52) {
		tmp = fabs(((cos(t) * eh) * t_2));
	} else if (t <= 1.75e-6) {
		tmp = fabs(fma((t_2 * cos(t)), eh, ((t * ew) / sqrt(fma(t_1, t_1, 1.0)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(t * ew))
	t_2 = sin(atan(Float64(eh / Float64(tan(t) * ew))))
	t_3 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (t <= -2.6e+173)
		tmp = t_3;
	elseif (t <= -1.4e+52)
		tmp = abs(Float64(Float64(cos(t) * eh) * t_2));
	elseif (t <= 1.75e-6)
		tmp = abs(fma(Float64(t_2 * cos(t)), eh, Float64(Float64(t * ew) / sqrt(fma(t_1, t_1, 1.0)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[ArcTan[N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -2.6e+173], t$95$3, If[LessEqual[t, -1.4e+52], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.75e-6], N[Abs[N[(N[(t$95$2 * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh + N[(N[(t * ew), $MachinePrecision] / N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.5999999999999999e173 or 1.74999999999999997e-6 < t

   1. Initial program 99.7%

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in ew around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

      3. lower-sin.f64 62.9

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t} \cdot ew}\right|} \]

   7. Applied rewrites 62.9%

      \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

   9. Applied rewrites 63.0%

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

   if -2.5999999999999999e173 < t < -1.4e52

   1. Initial program 99.4%

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

   4. Applied rewrites 83.5%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 64.6

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

   7. Applied rewrites 64.6%

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

   9. Applied rewrites 64.6%

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

   if -1.4e52 < t < 1.74999999999999997e-6

   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 99.9

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

   5. Applied rewrites 99.9%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot t}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   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 99.9%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 96.1

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

   10. Applied rewrites 96.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{+173}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{+52}:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right|\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\left|\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right) \cdot \cos t, eh, \frac{t \cdot ew}{\sqrt{\mathsf{fma}\left(\frac{eh}{t \cdot ew}, \frac{eh}{t \cdot ew}, 1\right)}}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right|\\ \mathbf{if}\;eh \leq -4.8 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 1.12 \cdot 10^{-127}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (* (cos t) eh) (sin (atan (/ eh (* (tan t) ew))))))))
   (if (<= eh -4.8e-73) t_1 (if (<= eh 1.12e-127) (fabs (* (sin t) ew)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((cos(t) * eh) * sin(atan((eh / (tan(t) * ew))))));
	double tmp;
	if (eh <= -4.8e-73) {
		tmp = t_1;
	} else if (eh <= 1.12e-127) {
		tmp = fabs((sin(t) * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(((cos(t) * eh) * sin(atan((eh / (tan(t) * ew))))))
    if (eh <= (-4.8d-73)) then
        tmp = t_1
    else if (eh <= 1.12d-127) then
        tmp = abs((sin(t) * ew))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs(((Math.cos(t) * eh) * Math.sin(Math.atan((eh / (Math.tan(t) * ew))))));
	double tmp;
	if (eh <= -4.8e-73) {
		tmp = t_1;
	} else if (eh <= 1.12e-127) {
		tmp = Math.abs((Math.sin(t) * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs(((math.cos(t) * eh) * math.sin(math.atan((eh / (math.tan(t) * ew))))))
	tmp = 0
	if eh <= -4.8e-73:
		tmp = t_1
	elif eh <= 1.12e-127:
		tmp = math.fabs((math.sin(t) * ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(cos(t) * eh) * sin(atan(Float64(eh / Float64(tan(t) * ew))))))
	tmp = 0.0
	if (eh <= -4.8e-73)
		tmp = t_1;
	elseif (eh <= 1.12e-127)
		tmp = abs(Float64(sin(t) * ew));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs(((cos(t) * eh) * sin(atan((eh / (tan(t) * ew))))));
	tmp = 0.0;
	if (eh <= -4.8e-73)
		tmp = t_1;
	elseif (eh <= 1.12e-127)
		tmp = abs((sin(t) * ew));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -4.8e-73], t$95$1, If[LessEqual[eh, 1.12e-127], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.80000000000000011e-73 or 1.1199999999999999e-127 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 78.4

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

   7. Applied rewrites 78.4%

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

   9. Applied rewrites 78.6%

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

   if -4.80000000000000011e-73 < eh < 1.1199999999999999e-127

   1. Initial program 99.8%

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

   4. Applied rewrites 74.7%

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

   5. Taylor expanded in ew around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

      3. lower-sin.f64 75.3

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t} \cdot ew}\right|} \]

   7. Applied rewrites 75.3%

      \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

   9. Applied rewrites 75.4%

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

Alternative 6: 69.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ t_2 := \frac{1}{\left|\frac{1}{\sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot t\_1}\right|}\\ t_3 := \frac{1}{\left|\frac{1}{\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333 \cdot \frac{eh}{ew}, t \cdot t, \frac{eh}{ew}\right)}{t}\right) \cdot t\_1}\right|}\\ \mathbf{if}\;eh \leq -2.2 \cdot 10^{+141}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq -4.8 \cdot 10^{-73}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eh \leq 1.12 \cdot 10^{-127}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{elif}\;eh \leq 1.9 \cdot 10^{+60}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh))
        (t_2 (/ 1.0 (fabs (/ 1.0 (* (sin (atan (/ eh (* t ew)))) t_1)))))
        (t_3
         (/
          1.0
          (fabs
           (/
            1.0
            (*
             (sin
              (atan
               (/
                (fma (* -0.3333333333333333 (/ eh ew)) (* t t) (/ eh ew))
                t)))
             t_1))))))
   (if (<= eh -2.2e+141)
     t_2
     (if (<= eh -4.8e-73)
       t_3
       (if (<= eh 1.12e-127)
         (fabs (* (sin t) ew))
         (if (<= eh 1.9e+60) t_3 t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double t_2 = 1.0 / fabs((1.0 / (sin(atan((eh / (t * ew)))) * t_1)));
	double t_3 = 1.0 / fabs((1.0 / (sin(atan((fma((-0.3333333333333333 * (eh / ew)), (t * t), (eh / ew)) / t))) * t_1)));
	double tmp;
	if (eh <= -2.2e+141) {
		tmp = t_2;
	} else if (eh <= -4.8e-73) {
		tmp = t_3;
	} else if (eh <= 1.12e-127) {
		tmp = fabs((sin(t) * ew));
	} else if (eh <= 1.9e+60) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	t_2 = Float64(1.0 / abs(Float64(1.0 / Float64(sin(atan(Float64(eh / Float64(t * ew)))) * t_1))))
	t_3 = Float64(1.0 / abs(Float64(1.0 / Float64(sin(atan(Float64(fma(Float64(-0.3333333333333333 * Float64(eh / ew)), Float64(t * t), Float64(eh / ew)) / t))) * t_1))))
	tmp = 0.0
	if (eh <= -2.2e+141)
		tmp = t_2;
	elseif (eh <= -4.8e-73)
		tmp = t_3;
	elseif (eh <= 1.12e-127)
		tmp = abs(Float64(sin(t) * ew));
	elseif (eh <= 1.9e+60)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Abs[N[(1.0 / N[(N[Sin[N[ArcTan[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[Abs[N[(1.0 / N[(N[Sin[N[ArcTan[N[(N[(N[(-0.3333333333333333 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -2.2e+141], t$95$2, If[LessEqual[eh, -4.8e-73], t$95$3, If[LessEqual[eh, 1.12e-127], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 1.9e+60], t$95$3, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -2.2e141 or 1.90000000000000005e60 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 85.5

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 75.6%

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

   if -2.2e141 < eh < -4.80000000000000011e-73 or 1.1199999999999999e-127 < eh < 1.90000000000000005e60

   1. Initial program 99.8%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 71.3

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

   7. Applied rewrites 71.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 70.3%

       \[\leadsto \frac{1}{\left|\frac{1}{\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333 \cdot \frac{eh}{ew}, t \cdot t, \frac{eh}{ew}\right)}{t}\right)}\right|} \]

   if -4.80000000000000011e-73 < eh < 1.1199999999999999e-127

   1. Initial program 99.8%

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

   4. Applied rewrites 74.7%

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

   5. Taylor expanded in ew around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

      3. lower-sin.f64 75.3

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t} \cdot ew}\right|} \]

   7. Applied rewrites 75.3%

      \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

   9. Applied rewrites 75.4%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot \left(\cos t \cdot eh\right)}\right|}\\ \mathbf{elif}\;eh \leq -4.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333 \cdot \frac{eh}{ew}, t \cdot t, \frac{eh}{ew}\right)}{t}\right) \cdot \left(\cos t \cdot eh\right)}\right|}\\ \mathbf{elif}\;eh \leq 1.12 \cdot 10^{-127}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{elif}\;eh \leq 1.9 \cdot 10^{+60}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333 \cdot \frac{eh}{ew}, t \cdot t, \frac{eh}{ew}\right)}{t}\right) \cdot \left(\cos t \cdot eh\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot \left(\cos t \cdot eh\right)}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\left|\frac{1}{\sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot \left(\cos t \cdot eh\right)}\right|}\\ \mathbf{if}\;eh \leq -9 \cdot 10^{+28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 1.12 \cdot 10^{-127}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (/
          1.0
          (fabs (/ 1.0 (* (sin (atan (/ eh (* t ew)))) (* (cos t) eh)))))))
   (if (<= eh -9e+28) t_1 (if (<= eh 1.12e-127) (fabs (* (sin t) ew)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = 1.0 / fabs((1.0 / (sin(atan((eh / (t * ew)))) * (cos(t) * eh))));
	double tmp;
	if (eh <= -9e+28) {
		tmp = t_1;
	} else if (eh <= 1.12e-127) {
		tmp = fabs((sin(t) * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / abs((1.0d0 / (sin(atan((eh / (t * ew)))) * (cos(t) * eh))))
    if (eh <= (-9d+28)) then
        tmp = t_1
    else if (eh <= 1.12d-127) then
        tmp = abs((sin(t) * ew))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = 1.0 / Math.abs((1.0 / (Math.sin(Math.atan((eh / (t * ew)))) * (Math.cos(t) * eh))));
	double tmp;
	if (eh <= -9e+28) {
		tmp = t_1;
	} else if (eh <= 1.12e-127) {
		tmp = Math.abs((Math.sin(t) * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = 1.0 / math.fabs((1.0 / (math.sin(math.atan((eh / (t * ew)))) * (math.cos(t) * eh))))
	tmp = 0
	if eh <= -9e+28:
		tmp = t_1
	elif eh <= 1.12e-127:
		tmp = math.fabs((math.sin(t) * ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(1.0 / abs(Float64(1.0 / Float64(sin(atan(Float64(eh / Float64(t * ew)))) * Float64(cos(t) * eh)))))
	tmp = 0.0
	if (eh <= -9e+28)
		tmp = t_1;
	elseif (eh <= 1.12e-127)
		tmp = abs(Float64(sin(t) * ew));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = 1.0 / abs((1.0 / (sin(atan((eh / (t * ew)))) * (cos(t) * eh))));
	tmp = 0.0;
	if (eh <= -9e+28)
		tmp = t_1;
	elseif (eh <= 1.12e-127)
		tmp = abs((sin(t) * ew));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(1.0 / N[Abs[N[(1.0 / N[(N[Sin[N[ArcTan[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -9e+28], t$95$1, If[LessEqual[eh, 1.12e-127], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.9999999999999994e28 or 1.1199999999999999e-127 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 93.2%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 81.8

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

   7. Applied rewrites 81.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 68.4%

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

   if -8.9999999999999994e28 < eh < 1.1199999999999999e-127

   1. Initial program 99.8%

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in ew around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

      3. lower-sin.f64 69.9

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t} \cdot ew}\right|} \]

   7. Applied rewrites 69.9%

      \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

   9. Applied rewrites 70.0%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -9 \cdot 10^{+28}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot \left(\cos t \cdot eh\right)}\right|}\\ \mathbf{elif}\;eh \leq 1.12 \cdot 10^{-127}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot \left(\cos t \cdot eh\right)}\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;t \leq -1 \cdot 10^{-36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-7}:\\ \;\;\;\;\left|-eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= t -1e-36) t_1 (if (<= t 3.3e-7) (fabs (- eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (t <= -1e-36) {
		tmp = t_1;
	} else if (t <= 3.3e-7) {
		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 (t <= (-1d-36)) then
        tmp = t_1
    else if (t <= 3.3d-7) 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 (t <= -1e-36) {
		tmp = t_1;
	} else if (t <= 3.3e-7) {
		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 t <= -1e-36:
		tmp = t_1
	elif t <= 3.3e-7:
		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 (t <= -1e-36)
		tmp = t_1;
	elseif (t <= 3.3e-7)
		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 (t <= -1e-36)
		tmp = t_1;
	elseif (t <= 3.3e-7)
		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[t, -1e-36], t$95$1, If[LessEqual[t, 3.3e-7], N[Abs[(-eh)], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -9.9999999999999994e-37 or 3.3000000000000002e-7 < t

   1. Initial program 99.6%

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

   4. Applied rewrites 90.2%

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

   5. Taylor expanded in ew around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

      3. lower-sin.f64 56.7

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t} \cdot ew}\right|} \]

   7. Applied rewrites 56.7%

      \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

   9. Applied rewrites 56.7%

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

   if -9.9999999999999994e-37 < t < 3.3000000000000002e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. 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. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-cos.f64 79.3

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

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 10.3%

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

   8. Taylor expanded in eh around -inf

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

   10. Applied rewrites 79.6%

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

Alternative 9: 49.2% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot ew\\ \mathbf{if}\;t \leq -4.4 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.6 \cdot 10^{-7}:\\ \;\;\;\;\left|-eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) ew)))
   (if (<= t -4.4e+47) t_1 (if (<= t 8.6e-7) (fabs (- eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * ew;
	double tmp;
	if (t <= -4.4e+47) {
		tmp = t_1;
	} else if (t <= 8.6e-7) {
		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 = sin(t) * ew
    if (t <= (-4.4d+47)) then
        tmp = t_1
    else if (t <= 8.6d-7) 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.sin(t) * ew;
	double tmp;
	if (t <= -4.4e+47) {
		tmp = t_1;
	} else if (t <= 8.6e-7) {
		tmp = Math.abs(-eh);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(t) * ew
	tmp = 0
	if t <= -4.4e+47:
		tmp = t_1
	elif t <= 8.6e-7:
		tmp = math.fabs(-eh)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * ew)
	tmp = 0.0
	if (t <= -4.4e+47)
		tmp = t_1;
	elseif (t <= 8.6e-7)
		tmp = abs(Float64(-eh));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(t) * ew;
	tmp = 0.0;
	if (t <= -4.4e+47)
		tmp = t_1;
	elseif (t <= 8.6e-7)
		tmp = abs(-eh);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]}, If[LessEqual[t, -4.4e+47], t$95$1, If[LessEqual[t, 8.6e-7], N[Abs[(-eh)], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.3999999999999999e47 or 8.6000000000000002e-7 < t

   1. Initial program 99.6%

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

   4. Applied rewrites 89.6%

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

   5. Taylor expanded in ew around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

      3. lower-sin.f64 57.8

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t} \cdot ew}\right|} \]

   7. Applied rewrites 57.8%

      \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

   9. Applied rewrites 25.2%

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

   if -4.3999999999999999e47 < t < 8.6000000000000002e-7

   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. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-cos.f64 71.6

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

   5. Applied rewrites 71.6%

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

   7. Applied rewrites 10.6%

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

   8. Taylor expanded in eh around -inf

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

   10. Applied rewrites 72.0%

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

Alternative 10: 45.3% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|-eh\right|\\ \mathbf{if}\;eh \leq -1.15 \cdot 10^{-129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 6.8 \cdot 10^{-229}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{t \cdot ew}\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (- eh))))
   (if (<= eh -1.15e-129)
     t_1
     (if (<= eh 6.8e-229) (/ 1.0 (fabs (/ 1.0 (* t ew)))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(-eh);
	double tmp;
	if (eh <= -1.15e-129) {
		tmp = t_1;
	} else if (eh <= 6.8e-229) {
		tmp = 1.0 / fabs((1.0 / (t * ew)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(-eh)
    if (eh <= (-1.15d-129)) then
        tmp = t_1
    else if (eh <= 6.8d-229) then
        tmp = 1.0d0 / abs((1.0d0 / (t * ew)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs(-eh);
	double tmp;
	if (eh <= -1.15e-129) {
		tmp = t_1;
	} else if (eh <= 6.8e-229) {
		tmp = 1.0 / Math.abs((1.0 / (t * ew)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs(-eh)
	tmp = 0
	if eh <= -1.15e-129:
		tmp = t_1
	elif eh <= 6.8e-229:
		tmp = 1.0 / math.fabs((1.0 / (t * ew)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(-eh))
	tmp = 0.0
	if (eh <= -1.15e-129)
		tmp = t_1;
	elseif (eh <= 6.8e-229)
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(t * ew))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs(-eh);
	tmp = 0.0;
	if (eh <= -1.15e-129)
		tmp = t_1;
	elseif (eh <= 6.8e-229)
		tmp = 1.0 / abs((1.0 / (t * ew)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[(-eh)], $MachinePrecision]}, If[LessEqual[eh, -1.15e-129], t$95$1, If[LessEqual[eh, 6.8e-229], N[(1.0 / N[Abs[N[(1.0 / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.15e-129 or 6.7999999999999998e-229 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. 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. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-cos.f64 50.4

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

   5. Applied rewrites 50.4%

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

   7. Applied rewrites 8.9%

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

   8. Taylor expanded in eh around -inf

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

   10. Applied rewrites 50.9%

       \[\leadsto \left|-eh\right| \]

   if -1.15e-129 < eh < 6.7999999999999998e-229

   1. Initial program 99.7%

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in ew around inf

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

      3. lower-sin.f64 85.2

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t} \cdot ew}\right|} \]

   7. Applied rewrites 85.2%

      \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{\sin t \cdot ew}}\right|} \]

   8. Taylor expanded in t around 0

      \[\leadsto \frac{1}{\left|\frac{1}{ew \cdot \color{blue}{t}}\right|} \]
   9. Step-by-step derivation

   10. Applied rewrites 35.7%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.15 \cdot 10^{-129}:\\ \;\;\;\;\left|-eh\right|\\ \mathbf{elif}\;eh \leq 6.8 \cdot 10^{-229}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{t \cdot ew}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|-eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.9% 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 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
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. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-cos.f64 43.4

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

5. Applied rewrites 43.4%

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

7. Applied rewrites 7.6%

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

8. Taylor expanded in eh around -inf

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

10. Applied rewrites 43.9%

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

Reproduce

herbie shell --seed 2024235 
(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))))))))