Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 20.1s

Alternatives: 10

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

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

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]]

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. Add Preprocessing

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.2

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

5. Applied rewrites 99.2%

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

6. Final simplification 99.2%

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

Alternative 3: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 88.5%

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

5. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-atan.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-tan.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 98.8

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

7. Applied rewrites 98.8%

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

Alternative 4: 93.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \mathsf{fma}\left(\cos t, eh \cdot \frac{\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, eh \cdot \frac{t \cdot t}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}, \sin t\right)\right|\\ \mathbf{if}\;ew \leq -5.3 \cdot 10^{-157}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 3.2 \cdot 10^{-47}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           ew
           (fma
            (cos t)
            (*
             eh
             (/
              (sin
               (atan
                (/
                 (fma -0.3333333333333333 (* eh (/ (* t t) ew)) (/ eh ew))
                 t)))
              ew))
            (sin t))))))
   (if (<= ew -5.3e-157)
     t_1
     (if (<= ew 3.2e-47)
       (fabs (* (* eh (cos t)) (sin (atan (/ eh (* ew (tan t)))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * fma(cos(t), (eh * (sin(atan((fma(-0.3333333333333333, (eh * ((t * t) / ew)), (eh / ew)) / t))) / ew)), sin(t))));
	double tmp;
	if (ew <= -5.3e-157) {
		tmp = t_1;
	} else if (ew <= 3.2e-47) {
		tmp = fabs(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * fma(cos(t), Float64(eh * Float64(sin(atan(Float64(fma(-0.3333333333333333, Float64(eh * Float64(Float64(t * t) / ew)), Float64(eh / ew)) / t))) / ew)), sin(t))))
	tmp = 0.0
	if (ew <= -5.3e-157)
		tmp = t_1;
	elseif (ew <= 3.2e-47)
		tmp = abs(Float64(Float64(eh * cos(t)) * sin(atan(Float64(eh / Float64(ew * tan(t)))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] * N[(eh * N[(N[Sin[N[ArcTan[N[(N[(-0.3333333333333333 * N[(eh * N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] + N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -5.3e-157], t$95$1, If[LessEqual[ew, 3.2e-47], N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.3000000000000002e-157 or 3.1999999999999999e-47 < 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 89.1%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 97.2%

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

      1. lift-cos.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-sin.f64 N/A

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

   9. Applied rewrites 97.2%

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

   10. Taylor expanded in t around 0

       \[\leadsto \left|ew \cdot \mathsf{fma}\left(\cos t, \frac{\sin \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}}{ew} \cdot eh, \sin t\right)\right| \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 96.9

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

   12. Applied rewrites 96.9%

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

   if -5.3000000000000002e-157 < ew < 3.1999999999999999e-47

   1. Initial program 99.9%

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

   3. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-tan.f64 94.0

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

   5. Applied rewrites 94.0%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5.3 \cdot 10^{-157}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(\cos t, eh \cdot \frac{\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, eh \cdot \frac{t \cdot t}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}, \sin t\right)\right|\\ \mathbf{elif}\;ew \leq 3.2 \cdot 10^{-47}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(\cos t, eh \cdot \frac{\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, eh \cdot \frac{t \cdot t}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}, \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \mathsf{fma}\left(\cos t, eh \cdot \frac{\sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)}{ew}, \sin t\right)\right|\\ \mathbf{if}\;ew \leq -3.8 \cdot 10^{-38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 8.5 \cdot 10^{-8}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           ew
           (fma (cos t) (* eh (/ (sin (atan (/ eh (* ew t)))) ew)) (sin t))))))
   (if (<= ew -3.8e-38)
     t_1
     (if (<= ew 8.5e-8)
       (fabs (* (* eh (cos t)) (sin (atan (/ eh (* ew (tan t)))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * fma(cos(t), (eh * (sin(atan((eh / (ew * t)))) / ew)), sin(t))));
	double tmp;
	if (ew <= -3.8e-38) {
		tmp = t_1;
	} else if (ew <= 8.5e-8) {
		tmp = fabs(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * fma(cos(t), Float64(eh * Float64(sin(atan(Float64(eh / Float64(ew * t)))) / ew)), sin(t))))
	tmp = 0.0
	if (ew <= -3.8e-38)
		tmp = t_1;
	elseif (ew <= 8.5e-8)
		tmp = abs(Float64(Float64(eh * cos(t)) * sin(atan(Float64(eh / Float64(ew * tan(t)))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] * N[(eh * N[(N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] + N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -3.8e-38], t$95$1, If[LessEqual[ew, 8.5e-8], N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.8e-38 or 8.49999999999999935e-8 < 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 89.3%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 98.2%

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

      1. lift-cos.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-sin.f64 N/A

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

   9. Applied rewrites 98.1%

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

   10. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 93.4

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

   12. Applied rewrites 93.4%

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

   if -3.8e-38 < ew < 8.49999999999999935e-8

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-tan.f64 91.0

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

   5. Applied rewrites 91.0%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.8 \cdot 10^{-38}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(\cos t, eh \cdot \frac{\sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)}{ew}, \sin t\right)\right|\\ \mathbf{elif}\;ew \leq 8.5 \cdot 10^{-8}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(\cos t, eh \cdot \frac{\sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)}{ew}, \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;ew \leq -7.5 \cdot 10^{+105}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 2.6 \cdot 10^{+91}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= ew -7.5e+105)
     t_1
     (if (<= ew 2.6e+91)
       (fabs (* (* eh (cos t)) (sin (atan (/ eh (* ew (tan t)))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (ew <= -7.5e+105) {
		tmp = t_1;
	} else if (ew <= 2.6e+91) {
		tmp = fabs(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))));
	} 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((ew * sin(t)))
    if (ew <= (-7.5d+105)) then
        tmp = t_1
    else if (ew <= 2.6d+91) then
        tmp = abs(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))))
    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((ew * Math.sin(t)));
	double tmp;
	if (ew <= -7.5e+105) {
		tmp = t_1;
	} else if (ew <= 2.6e+91) {
		tmp = Math.abs(((eh * Math.cos(t)) * Math.sin(Math.atan((eh / (ew * Math.tan(t)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.sin(t)))
	tmp = 0
	if ew <= -7.5e+105:
		tmp = t_1
	elif ew <= 2.6e+91:
		tmp = math.fabs(((eh * math.cos(t)) * math.sin(math.atan((eh / (ew * math.tan(t)))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (ew <= -7.5e+105)
		tmp = t_1;
	elseif (ew <= 2.6e+91)
		tmp = abs(Float64(Float64(eh * cos(t)) * sin(atan(Float64(eh / Float64(ew * tan(t)))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * sin(t)));
	tmp = 0.0;
	if (ew <= -7.5e+105)
		tmp = t_1;
	elseif (ew <= 2.6e+91)
		tmp = abs(((eh * cos(t)) * sin(atan((eh / (ew * tan(t)))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -7.5e+105], t$95$1, If[LessEqual[ew, 2.6e+91], N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -7.5000000000000002e105 or 2.6e91 < 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 86.0%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 74.9

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

   7. Applied rewrites 74.9%

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

   if -7.5000000000000002e105 < ew < 2.6e91

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-tan.f64 81.7

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

   5. Applied rewrites 81.7%

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

Alternative 7: 69.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;t \leq -900000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 0.00019:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}{ew}, t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= t -900000000.0)
     t_1
     (if (<= t 0.00019)
       (fabs (* ew (fma eh (/ (sin (atan (/ eh (* ew (tan t))))) ew) t)))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (t <= -900000000.0) {
		tmp = t_1;
	} else if (t <= 0.00019) {
		tmp = fabs((ew * fma(eh, (sin(atan((eh / (ew * tan(t))))) / ew), t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (t <= -900000000.0)
		tmp = t_1;
	elseif (t <= 0.00019)
		tmp = abs(Float64(ew * fma(eh, Float64(sin(atan(Float64(eh / Float64(ew * tan(t))))) / ew), t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -900000000.0], t$95$1, If[LessEqual[t, 0.00019], N[Abs[N[(ew * N[(eh * N[(N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / ew), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -9e8 or 1.9000000000000001e-4 < t

   1. Initial program 99.6%

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

   4. Applied rewrites 89.5%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 48.2

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

   7. Applied rewrites 48.2%

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

   if -9e8 < t < 1.9000000000000001e-4

   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 87.7%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 86.3%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-tan.f64 83.2

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

   10. Applied rewrites 83.2%

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

Alternative 8: 61.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;t \leq -900000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= t -900000000.0) t_1 (if (<= t 2.2e-6) (fabs eh) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (t <= -900000000.0) {
		tmp = t_1;
	} else if (t <= 2.2e-6) {
		tmp = fabs(eh);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((ew * sin(t)))
    if (t <= (-900000000.0d0)) then
        tmp = t_1
    else if (t <= 2.2d-6) then
        tmp = abs(eh)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * Math.sin(t)));
	double tmp;
	if (t <= -900000000.0) {
		tmp = t_1;
	} else if (t <= 2.2e-6) {
		tmp = Math.abs(eh);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.sin(t)))
	tmp = 0
	if t <= -900000000.0:
		tmp = t_1
	elif t <= 2.2e-6:
		tmp = math.fabs(eh)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (t <= -900000000.0)
		tmp = t_1;
	elseif (t <= 2.2e-6)
		tmp = abs(eh);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * sin(t)));
	tmp = 0.0;
	if (t <= -900000000.0)
		tmp = t_1;
	elseif (t <= 2.2e-6)
		tmp = abs(eh);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -900000000.0], t$95$1, If[LessEqual[t, 2.2e-6], N[Abs[eh], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -9e8 or 2.2000000000000001e-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 89.5%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 48.2

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

   7. Applied rewrites 48.2%

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

   if -9e8 < t < 2.2000000000000001e-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. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-tan.f64 72.8

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

   5. Applied rewrites 72.8%

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

   7. Applied rewrites 11.5%

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

   8. Taylor expanded in eh around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \left|\color{blue}{\mathsf{neg}\left(eh\right)}\right| \]

      2. lower-neg.f64 73.1

         \[\leadsto \left|\color{blue}{-eh}\right| \]

   10. Applied rewrites 73.1%

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

       1. fabs-neg N/A

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

       2. lower-fabs.f64 73.1

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

   12. Applied rewrites 73.1%

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

Alternative 9: 45.6% accurate, 43.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -5.1 \cdot 10^{-202}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{elif}\;eh \leq 3.8 \cdot 10^{-148}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -5.1e-202)
   (fabs eh)
   (if (<= eh 3.8e-148) (fabs (* ew t)) (fabs eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -5.1e-202) {
		tmp = fabs(eh);
	} else if (eh <= 3.8e-148) {
		tmp = fabs((ew * t));
	} else {
		tmp = fabs(eh);
	}
	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) :: tmp
    if (eh <= (-5.1d-202)) then
        tmp = abs(eh)
    else if (eh <= 3.8d-148) then
        tmp = abs((ew * t))
    else
        tmp = abs(eh)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -5.1e-202) {
		tmp = Math.abs(eh);
	} else if (eh <= 3.8e-148) {
		tmp = Math.abs((ew * t));
	} else {
		tmp = Math.abs(eh);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -5.1e-202:
		tmp = math.fabs(eh)
	elif eh <= 3.8e-148:
		tmp = math.fabs((ew * t))
	else:
		tmp = math.fabs(eh)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -5.1e-202)
		tmp = abs(eh);
	elseif (eh <= 3.8e-148)
		tmp = abs(Float64(ew * t));
	else
		tmp = abs(eh);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -5.1e-202)
		tmp = abs(eh);
	elseif (eh <= 3.8e-148)
		tmp = abs((ew * t));
	else
		tmp = abs(eh);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -5.1e-202], N[Abs[eh], $MachinePrecision], If[LessEqual[eh, 3.8e-148], N[Abs[N[(ew * t), $MachinePrecision]], $MachinePrecision], N[Abs[eh], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.09999999999999996e-202 or 3.80000000000000014e-148 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-tan.f64 53.9

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

   5. Applied rewrites 53.9%

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

   7. Applied rewrites 10.3%

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

   8. Taylor expanded in eh around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \left|\color{blue}{\mathsf{neg}\left(eh\right)}\right| \]

      2. lower-neg.f64 54.2

         \[\leadsto \left|\color{blue}{-eh}\right| \]

   10. Applied rewrites 54.2%

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

       1. fabs-neg N/A

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

       2. lower-fabs.f64 54.2

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

   12. Applied rewrites 54.2%

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

   if -5.09999999999999996e-202 < eh < 3.80000000000000014e-148

   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 72.5%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 85.2

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

   7. Applied rewrites 85.2%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 42.4

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

   10. Applied rewrites 42.4%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -5.1 \cdot 10^{-202}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{elif}\;eh \leq 3.8 \cdot 10^{-148}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.8% accurate, 290.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-atan.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-tan.f64 47.5

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

5. Applied rewrites 47.5%

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

7. Applied rewrites 9.1%

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

8. Taylor expanded in eh around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \left|\color{blue}{\mathsf{neg}\left(eh\right)}\right| \]

   2. lower-neg.f64 47.9

      \[\leadsto \left|\color{blue}{-eh}\right| \]

10. Applied rewrites 47.9%

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

    1. fabs-neg N/A

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

    2. lower-fabs.f64 47.9

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

12. Applied rewrites 47.9%

    \[\leadsto \color{blue}{\left|eh\right|} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))))))))