Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.1s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 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]], $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. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 3: 99.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] + 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]], $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. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 99.4

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

9. Applied rewrites 99.4%

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

Alternative 4: 99.0% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* t ew))))
   (fabs
    (fma
     (cos t)
     (* (sin (atan (/ eh (* (tan t) ew)))) 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(cos(t), (sin(atan((eh / (tan(t) * ew)))) * 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(cos(t), Float64(sin(atan(Float64(eh / Float64(tan(t) * ew)))) * 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[Cos[t], $MachinePrecision] * N[(N[Sin[N[ArcTan[N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision] + 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. 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. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 99.4

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

   9. lift-*.f64 N/A

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

9. Applied rewrites 99.4%

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

10. Final simplification 99.4%

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

Alternative 5: 98.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(eh, ew, t)
	return abs(fma(sin(t), ew, Float64(Float64(cos(t) * eh) * sin(atan(Float64(eh / Float64(tan(t) * ew)))))))
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[(N[Tan[t], $MachinePrecision] * ew), $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

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

   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 88.5%

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

5. Taylor expanded in ew around inf

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

   1. lower-sin.f64 98.8

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

7. Applied rewrites 98.8%

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

8. Final simplification 98.8%

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

Alternative 6: 91.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{t \cdot ew}\\ t_2 := \sin t \cdot ew\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+35}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{eh}{\tan t \cdot ew}, t\_2, \frac{t\_2}{1}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{1}{\sqrt{\mathsf{fma}\left(t\_1, t\_1, 1\right)}}, t\_2, \sin \tan^{-1} t\_1 \cdot \left(\cos t \cdot eh\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* t ew))) (t_2 (* (sin t) ew)))
   (if (<= t -1.8e+35)
     (fabs (fma (/ eh (* (tan t) ew)) t_2 (/ t_2 1.0)))
     (fabs
      (fma
       (/ 1.0 (sqrt (fma t_1 t_1 1.0)))
       t_2
       (* (sin (atan t_1)) (* (cos t) eh)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (t * ew);
	double t_2 = sin(t) * ew;
	double tmp;
	if (t <= -1.8e+35) {
		tmp = fabs(fma((eh / (tan(t) * ew)), t_2, (t_2 / 1.0)));
	} else {
		tmp = fabs(fma((1.0 / sqrt(fma(t_1, t_1, 1.0))), t_2, (sin(atan(t_1)) * (cos(t) * eh))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(t * ew))
	t_2 = Float64(sin(t) * ew)
	tmp = 0.0
	if (t <= -1.8e+35)
		tmp = abs(fma(Float64(eh / Float64(tan(t) * ew)), t_2, Float64(t_2 / 1.0)));
	else
		tmp = abs(fma(Float64(1.0 / sqrt(fma(t_1, t_1, 1.0))), t_2, Float64(sin(atan(t_1)) * Float64(cos(t) * eh))));
	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[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]}, If[LessEqual[t, -1.8e+35], N[Abs[N[(N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(t$95$2 / 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(1.0 / N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8e35

   1. Initial program 99.5%

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

   4. Applied rewrites 53.7%

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

   5. Taylor expanded in ew around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 73.8

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

   7. Applied rewrites 73.8%

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

   8. Taylor expanded in ew around inf

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

   10. Applied rewrites 88.6%

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

   if -1.8e35 < t

   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

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.7

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in t around 0

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

       1. lower-*.f64 94.6

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

   12. Applied rewrites 94.6%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+35}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{eh}{\tan t \cdot ew}, \sin t \cdot ew, \frac{\sin t \cdot ew}{1}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{1}{\sqrt{\mathsf{fma}\left(\frac{eh}{t \cdot ew}, \frac{eh}{t \cdot ew}, 1\right)}}, \sin t \cdot ew, \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot \left(\cos t \cdot eh\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot ew\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+35}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{eh}{\tan t \cdot ew}, t\_1, \frac{t\_1}{1}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\sin t, ew, \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot \left(\cos t \cdot eh\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) ew)))
   (if (<= t -1.8e+35)
     (fabs (fma (/ eh (* (tan t) ew)) t_1 (/ t_1 1.0)))
     (fabs (fma (sin t) ew (* (sin (atan (/ eh (* t ew)))) (* (cos t) eh)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * ew;
	double tmp;
	if (t <= -1.8e+35) {
		tmp = fabs(fma((eh / (tan(t) * ew)), t_1, (t_1 / 1.0)));
	} else {
		tmp = fabs(fma(sin(t), ew, (sin(atan((eh / (t * ew)))) * (cos(t) * eh))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * ew)
	tmp = 0.0
	if (t <= -1.8e+35)
		tmp = abs(fma(Float64(eh / Float64(tan(t) * ew)), t_1, Float64(t_1 / 1.0)));
	else
		tmp = abs(fma(sin(t), ew, Float64(sin(atan(Float64(eh / Float64(t * ew)))) * Float64(cos(t) * eh))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]}, If[LessEqual[t, -1.8e+35], N[Abs[N[(N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(t$95$1 / 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew + N[(N[Sin[N[ArcTan[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8e35

   1. Initial program 99.5%

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

   4. Applied rewrites 53.7%

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

   5. Taylor expanded in ew around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 73.8

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

   7. Applied rewrites 73.8%

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

   8. Taylor expanded in ew around inf

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

   10. Applied rewrites 88.6%

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

   if -1.8e35 < t

   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 89.8%

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

   5. Taylor expanded in ew around inf

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

      1. lower-sin.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 93.8

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

   10. Applied rewrites 93.8%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+35}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{eh}{\tan t \cdot ew}, \sin t \cdot ew, \frac{\sin t \cdot ew}{1}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\sin t, ew, \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot \left(\cos t \cdot eh\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot ew\\ t_2 := \left|\mathsf{fma}\left(\frac{eh}{\tan t \cdot ew}, t\_1, \frac{t\_1}{1}\right)\right|\\ \mathbf{if}\;t \leq -155000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\left|\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, t \cdot t, 1\right) \cdot t, ew, \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot \left(\cos t \cdot eh\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) ew))
        (t_2 (fabs (fma (/ eh (* (tan t) ew)) t_1 (/ t_1 1.0)))))
   (if (<= t -155000000000.0)
     t_2
     (if (<= t 1.75e-6)
       (fabs
        (fma
         (* (fma -0.16666666666666666 (* t t) 1.0) t)
         ew
         (* (sin (atan (/ eh (* t ew)))) (* (cos t) eh))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * ew;
	double t_2 = fabs(fma((eh / (tan(t) * ew)), t_1, (t_1 / 1.0)));
	double tmp;
	if (t <= -155000000000.0) {
		tmp = t_2;
	} else if (t <= 1.75e-6) {
		tmp = fabs(fma((fma(-0.16666666666666666, (t * t), 1.0) * t), ew, (sin(atan((eh / (t * ew)))) * (cos(t) * eh))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * ew)
	t_2 = abs(fma(Float64(eh / Float64(tan(t) * ew)), t_1, Float64(t_1 / 1.0)))
	tmp = 0.0
	if (t <= -155000000000.0)
		tmp = t_2;
	elseif (t <= 1.75e-6)
		tmp = abs(fma(Float64(fma(-0.16666666666666666, Float64(t * t), 1.0) * t), ew, Float64(sin(atan(Float64(eh / Float64(t * ew)))) * Float64(cos(t) * eh))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(t$95$1 / 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -155000000000.0], t$95$2, If[LessEqual[t, 1.75e-6], N[Abs[N[(N[(N[(-0.16666666666666666 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision] * ew + N[(N[Sin[N[ArcTan[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.55e11 or 1.74999999999999997e-6 < 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 59.7%

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

   5. Taylor expanded in ew around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 75.3

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

   7. Applied rewrites 75.3%

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

   8. Taylor expanded in ew around inf

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

   10. Applied rewrites 86.8%

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

   if -1.55e11 < t < 1.74999999999999997e-6

   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. 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 89.6%

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

   5. Taylor expanded in ew around inf

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

      1. lower-sin.f64 98.9

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 98.9

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

   10. Applied rewrites 98.9%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 98.8%

       \[\leadsto \left|\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, t \cdot t, 1\right) \cdot \color{blue}{t}, ew, \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \left(\cos t \cdot eh\right)\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -155000000000:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{eh}{\tan t \cdot ew}, \sin t \cdot ew, \frac{\sin t \cdot ew}{1}\right)\right|\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\left|\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, t \cdot t, 1\right) \cdot t, ew, \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot \left(\cos t \cdot eh\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{eh}{\tan t \cdot ew}, \sin t \cdot ew, \frac{\sin t \cdot ew}{1}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot ew\\ t_2 := \left|\mathsf{fma}\left(\frac{eh}{\tan t \cdot ew}, t\_1, \frac{t\_1}{1}\right)\right|\\ \mathbf{if}\;ew \leq -11500:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 7.8 \cdot 10^{-46}:\\ \;\;\;\;\left|\cos t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) ew))
        (t_2 (fabs (fma (/ eh (* (tan t) ew)) t_1 (/ t_1 1.0)))))
   (if (<= ew -11500.0) t_2 (if (<= ew 7.8e-46) (fabs (* (cos t) eh)) t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * ew;
	double t_2 = fabs(fma((eh / (tan(t) * ew)), t_1, (t_1 / 1.0)));
	double tmp;
	if (ew <= -11500.0) {
		tmp = t_2;
	} else if (ew <= 7.8e-46) {
		tmp = fabs((cos(t) * eh));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * ew)
	t_2 = abs(fma(Float64(eh / Float64(tan(t) * ew)), t_1, Float64(t_1 / 1.0)))
	tmp = 0.0
	if (ew <= -11500.0)
		tmp = t_2;
	elseif (ew <= 7.8e-46)
		tmp = abs(Float64(cos(t) * eh));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(t$95$1 / 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -11500.0], t$95$2, If[LessEqual[ew, 7.8e-46], N[Abs[N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -11500 or 7.8000000000000005e-46 < ew

   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 74.3%

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

   5. Taylor expanded in ew around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 86.3

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

   7. Applied rewrites 86.3%

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

   8. Taylor expanded in ew around inf

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

   10. Applied rewrites 91.5%

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

   if -11500 < ew < 7.8000000000000005e-46

   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 15.2%

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

   5. Taylor expanded in ew around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 88.0

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

   7. Applied rewrites 88.0%

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

Alternative 10: 80.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ t_2 := \frac{eh}{t \cdot ew}\\ t_3 := \left|\mathsf{fma}\left(\frac{t\_2}{\sqrt{\mathsf{fma}\left(t\_2, t\_2, 1\right)}}, t\_1, \sin t \cdot ew\right)\right|\\ \mathbf{if}\;ew \leq -70000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;ew \leq 1.65 \cdot 10^{+56}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh))
        (t_2 (/ eh (* t ew)))
        (t_3 (fabs (fma (/ t_2 (sqrt (fma t_2 t_2 1.0))) t_1 (* (sin t) ew)))))
   (if (<= ew -70000000000.0) t_3 (if (<= ew 1.65e+56) (fabs t_1) t_3))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double t_2 = eh / (t * ew);
	double t_3 = fabs(fma((t_2 / sqrt(fma(t_2, t_2, 1.0))), t_1, (sin(t) * ew)));
	double tmp;
	if (ew <= -70000000000.0) {
		tmp = t_3;
	} else if (ew <= 1.65e+56) {
		tmp = fabs(t_1);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	t_2 = Float64(eh / Float64(t * ew))
	t_3 = abs(fma(Float64(t_2 / sqrt(fma(t_2, t_2, 1.0))), t_1, Float64(sin(t) * ew)))
	tmp = 0.0
	if (ew <= -70000000000.0)
		tmp = t_3;
	elseif (ew <= 1.65e+56)
		tmp = abs(t_1);
	else
		tmp = t_3;
	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[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(N[(t$95$2 / N[Sqrt[N[(t$95$2 * t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1 + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -70000000000.0], t$95$3, If[LessEqual[ew, 1.65e+56], N[Abs[t$95$1], $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -7e10 or 1.65000000000000001e56 < ew

   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 94.3%

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

   5. Taylor expanded in ew around inf

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

      1. lower-sin.f64 98.9

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 95.2

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

   10. Applied rewrites 95.2%

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

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lower-fma.f64 N/A

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

   12. Applied rewrites 86.6%

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

   if -7e10 < ew < 1.65000000000000001e56

   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 19.0%

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

   5. Taylor expanded in ew around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 86.4

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

   7. Applied rewrites 86.4%

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

Alternative 11: 74.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot ew\\ \mathbf{if}\;ew \leq -1.35 \cdot 10^{+32}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{elif}\;ew \leq 4.8 \cdot 10^{+56}:\\ \;\;\;\;\left|\cos t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{t\_1}\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) ew)))
   (if (<= ew -1.35e+32)
     (fabs t_1)
     (if (<= ew 4.8e+56) (fabs (* (cos t) eh)) (/ 1.0 (fabs (/ 1.0 t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * ew;
	double tmp;
	if (ew <= -1.35e+32) {
		tmp = fabs(t_1);
	} else if (ew <= 4.8e+56) {
		tmp = fabs((cos(t) * eh));
	} else {
		tmp = 1.0 / fabs((1.0 / 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 (ew <= (-1.35d+32)) then
        tmp = abs(t_1)
    else if (ew <= 4.8d+56) then
        tmp = abs((cos(t) * eh))
    else
        tmp = 1.0d0 / abs((1.0d0 / 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 (ew <= -1.35e+32) {
		tmp = Math.abs(t_1);
	} else if (ew <= 4.8e+56) {
		tmp = Math.abs((Math.cos(t) * eh));
	} else {
		tmp = 1.0 / Math.abs((1.0 / t_1));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(t) * ew
	tmp = 0
	if ew <= -1.35e+32:
		tmp = math.fabs(t_1)
	elif ew <= 4.8e+56:
		tmp = math.fabs((math.cos(t) * eh))
	else:
		tmp = 1.0 / math.fabs((1.0 / t_1))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * ew)
	tmp = 0.0
	if (ew <= -1.35e+32)
		tmp = abs(t_1);
	elseif (ew <= 4.8e+56)
		tmp = abs(Float64(cos(t) * eh));
	else
		tmp = Float64(1.0 / abs(Float64(1.0 / t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(t) * ew;
	tmp = 0.0;
	if (ew <= -1.35e+32)
		tmp = abs(t_1);
	elseif (ew <= 4.8e+56)
		tmp = abs((cos(t) * eh));
	else
		tmp = 1.0 / abs((1.0 / 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[ew, -1.35e+32], N[Abs[t$95$1], $MachinePrecision], If[LessEqual[ew, 4.8e+56], N[Abs[N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Abs[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.35000000000000006e32

   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 93.2%

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

   5. Taylor expanded in ew around inf

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
   6. 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 71.5

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

   7. Applied rewrites 71.5%

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

   if -1.35000000000000006e32 < ew < 4.80000000000000027e56

   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 22.5%

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

   5. Taylor expanded in ew around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 85.7

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

   7. Applied rewrites 85.7%

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

   if 4.80000000000000027e56 < ew

   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 94.6%

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

   5. Taylor expanded in ew around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 75.0

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

   7. Applied rewrites 75.0%

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

Alternative 12: 74.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;ew \leq -1.35 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.65 \cdot 10^{+56}:\\ \;\;\;\;\left|\cos t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= ew -1.35e+32) t_1 (if (<= ew 1.65e+56) (fabs (* (cos t) eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (ew <= -1.35e+32) {
		tmp = t_1;
	} else if (ew <= 1.65e+56) {
		tmp = fabs((cos(t) * 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 <= (-1.35d+32)) then
        tmp = t_1
    else if (ew <= 1.65d+56) then
        tmp = abs((cos(t) * 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 <= -1.35e+32) {
		tmp = t_1;
	} else if (ew <= 1.65e+56) {
		tmp = Math.abs((Math.cos(t) * 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 <= -1.35e+32:
		tmp = t_1
	elif ew <= 1.65e+56:
		tmp = math.fabs((math.cos(t) * 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 <= -1.35e+32)
		tmp = t_1;
	elseif (ew <= 1.65e+56)
		tmp = abs(Float64(cos(t) * 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 <= -1.35e+32)
		tmp = t_1;
	elseif (ew <= 1.65e+56)
		tmp = abs((cos(t) * 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, -1.35e+32], t$95$1, If[LessEqual[ew, 1.65e+56], N[Abs[N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.35000000000000006e32 or 1.65000000000000001e56 < ew

   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 94.0%

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

   5. Taylor expanded in ew around inf

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
   6. 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 73.2

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

   7. Applied rewrites 73.2%

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

   if -1.35000000000000006e32 < ew < 1.65000000000000001e56

   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 22.5%

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

   5. Taylor expanded in ew around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 85.7

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

   7. Applied rewrites 85.7%

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

Alternative 13: 63.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot ew\\ \mathbf{if}\;ew \leq -1.48 \cdot 10^{+200}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 7.6 \cdot 10^{+86}:\\ \;\;\;\;\left|\cos t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) ew)))
   (if (<= ew -1.48e+200) t_1 (if (<= ew 7.6e+86) (fabs (* (cos t) eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * ew;
	double tmp;
	if (ew <= -1.48e+200) {
		tmp = t_1;
	} else if (ew <= 7.6e+86) {
		tmp = fabs((cos(t) * 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 (ew <= (-1.48d+200)) then
        tmp = t_1
    else if (ew <= 7.6d+86) then
        tmp = abs((cos(t) * 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 (ew <= -1.48e+200) {
		tmp = t_1;
	} else if (ew <= 7.6e+86) {
		tmp = Math.abs((Math.cos(t) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(t) * ew
	tmp = 0
	if ew <= -1.48e+200:
		tmp = t_1
	elif ew <= 7.6e+86:
		tmp = math.fabs((math.cos(t) * eh))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * ew)
	tmp = 0.0
	if (ew <= -1.48e+200)
		tmp = t_1;
	elseif (ew <= 7.6e+86)
		tmp = abs(Float64(cos(t) * 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 (ew <= -1.48e+200)
		tmp = t_1;
	elseif (ew <= 7.6e+86)
		tmp = abs((cos(t) * 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[ew, -1.48e+200], t$95$1, If[LessEqual[ew, 7.6e+86], N[Abs[N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.48000000000000008e200 or 7.59999999999999956e86 < ew

   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 92.6%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

   6. Applied rewrites 45.3%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(\frac{ew}{\sqrt{\mathsf{fma}\left({\left(\tan t \cdot ew\right)}^{-2}, eh \cdot eh, 1\right)}}, \sin t, \left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)}^{0.5}\right)}^{2}} \]

   7. Taylor expanded in ew around inf

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 43.5

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

   9. Applied rewrites 43.5%

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

   if -1.48000000000000008e200 < ew < 7.59999999999999956e86

   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 35.9%

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

   5. Taylor expanded in ew around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 74.0

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

   7. Applied rewrites 74.0%

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

Alternative 14: 47.7% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot ew\\ \mathbf{if}\;t \leq -8.2 \cdot 10^{+67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-19}:\\ \;\;\;\;\left|1 \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) ew)))
   (if (<= t -8.2e+67) t_1 (if (<= t 1.45e-19) (fabs (* 1.0 eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * ew;
	double tmp;
	if (t <= -8.2e+67) {
		tmp = t_1;
	} else if (t <= 1.45e-19) {
		tmp = fabs((1.0 * 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 <= (-8.2d+67)) then
        tmp = t_1
    else if (t <= 1.45d-19) then
        tmp = abs((1.0d0 * 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 <= -8.2e+67) {
		tmp = t_1;
	} else if (t <= 1.45e-19) {
		tmp = Math.abs((1.0 * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(t) * ew
	tmp = 0
	if t <= -8.2e+67:
		tmp = t_1
	elif t <= 1.45e-19:
		tmp = math.fabs((1.0 * eh))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * ew)
	tmp = 0.0
	if (t <= -8.2e+67)
		tmp = t_1;
	elseif (t <= 1.45e-19)
		tmp = abs(Float64(1.0 * 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 <= -8.2e+67)
		tmp = t_1;
	elseif (t <= 1.45e-19)
		tmp = abs((1.0 * 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, -8.2e+67], t$95$1, If[LessEqual[t, 1.45e-19], N[Abs[N[(1.0 * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.19999999999999959e67 or 1.45e-19 < 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 86.8%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

   6. Applied rewrites 47.9%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(\frac{ew}{\sqrt{\mathsf{fma}\left({\left(\tan t \cdot ew\right)}^{-2}, eh \cdot eh, 1\right)}}, \sin t, \left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right)\right)}^{0.5}\right)}^{2}} \]

   7. Taylor expanded in ew around inf

      \[\leadsto \color{blue}{ew \cdot \sin t} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 33.4

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

   9. Applied rewrites 33.4%

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

   if -8.19999999999999959e67 < t < 1.45e-19

   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

   4. Applied rewrites 39.5%

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

   5. Taylor expanded in ew around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 73.6

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

   7. Applied rewrites 73.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 69.0%

       \[\leadsto \left|1 \cdot eh\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 42.3% accurate, 108.8× speedup

Math

\[\begin{array}{l} \\ \left|1 \cdot eh\right| \end{array} \]

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((1.0 * 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((1.0d0 * eh))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(1.0 * eh), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 49.1%

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

5. Taylor expanded in ew around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 60.1

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

7. Applied rewrites 60.1%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 43.1%

    \[\leadsto \left|1 \cdot eh\right| \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024240 
(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))))))))