Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 18.5s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / -ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 97.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := \left(-eh\right) \cdot t\_1\\ t_3 := ew \cdot \cos t\\ t_4 := \left|eh \cdot \left(\frac{t\_3}{eh} - \sin t \cdot \sin \tan^{-1} t\_2\right)\right|\\ \mathbf{if}\;eh \leq -2 \cdot 10^{-41}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eh \leq 0.46:\\ \;\;\;\;\left|\frac{t\_3 + eh \cdot \left(\sin t \cdot \left(eh \cdot t\_1\right)\right)}{\sqrt{1 + {t\_2}^{2}}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew))
        (t_2 (* (- eh) t_1))
        (t_3 (* ew (cos t)))
        (t_4 (fabs (* eh (- (/ t_3 eh) (* (sin t) (sin (atan t_2))))))))
   (if (<= eh -2e-41)
     t_4
     (if (<= eh 0.46)
       (fabs
        (/ (+ t_3 (* eh (* (sin t) (* eh t_1)))) (sqrt (+ 1.0 (pow t_2 2.0)))))
       t_4))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = -eh * t_1;
	double t_3 = ew * cos(t);
	double t_4 = fabs((eh * ((t_3 / eh) - (sin(t) * sin(atan(t_2))))));
	double tmp;
	if (eh <= -2e-41) {
		tmp = t_4;
	} else if (eh <= 0.46) {
		tmp = fabs(((t_3 + (eh * (sin(t) * (eh * t_1)))) / sqrt((1.0 + pow(t_2, 2.0)))));
	} else {
		tmp = t_4;
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = tan(t) / ew
    t_2 = -eh * t_1
    t_3 = ew * cos(t)
    t_4 = abs((eh * ((t_3 / eh) - (sin(t) * sin(atan(t_2))))))
    if (eh <= (-2d-41)) then
        tmp = t_4
    else if (eh <= 0.46d0) then
        tmp = abs(((t_3 + (eh * (sin(t) * (eh * t_1)))) / sqrt((1.0d0 + (t_2 ** 2.0d0)))))
    else
        tmp = t_4
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.tan(t) / ew;
	double t_2 = -eh * t_1;
	double t_3 = ew * Math.cos(t);
	double t_4 = Math.abs((eh * ((t_3 / eh) - (Math.sin(t) * Math.sin(Math.atan(t_2))))));
	double tmp;
	if (eh <= -2e-41) {
		tmp = t_4;
	} else if (eh <= 0.46) {
		tmp = Math.abs(((t_3 + (eh * (Math.sin(t) * (eh * t_1)))) / Math.sqrt((1.0 + Math.pow(t_2, 2.0)))));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.tan(t) / ew
	t_2 = -eh * t_1
	t_3 = ew * math.cos(t)
	t_4 = math.fabs((eh * ((t_3 / eh) - (math.sin(t) * math.sin(math.atan(t_2))))))
	tmp = 0
	if eh <= -2e-41:
		tmp = t_4
	elif eh <= 0.46:
		tmp = math.fabs(((t_3 + (eh * (math.sin(t) * (eh * t_1)))) / math.sqrt((1.0 + math.pow(t_2, 2.0)))))
	else:
		tmp = t_4
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = Float64(Float64(-eh) * t_1)
	t_3 = Float64(ew * cos(t))
	t_4 = abs(Float64(eh * Float64(Float64(t_3 / eh) - Float64(sin(t) * sin(atan(t_2))))))
	tmp = 0.0
	if (eh <= -2e-41)
		tmp = t_4;
	elseif (eh <= 0.46)
		tmp = abs(Float64(Float64(t_3 + Float64(eh * Float64(sin(t) * Float64(eh * t_1)))) / sqrt(Float64(1.0 + (t_2 ^ 2.0)))));
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = tan(t) / ew;
	t_2 = -eh * t_1;
	t_3 = ew * cos(t);
	t_4 = abs((eh * ((t_3 / eh) - (sin(t) * sin(atan(t_2))))));
	tmp = 0.0;
	if (eh <= -2e-41)
		tmp = t_4;
	elseif (eh <= 0.46)
		tmp = abs(((t_3 + (eh * (sin(t) * (eh * t_1)))) / sqrt((1.0 + (t_2 ^ 2.0)))));
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[((-eh) * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Abs[N[(eh * N[(N[(t$95$3 / eh), $MachinePrecision] - N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2e-41], t$95$4, If[LessEqual[eh, 0.46], N[Abs[N[(N[(t$95$3 + N[(eh * N[(N[Sin[t], $MachinePrecision] * N[(eh * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.00000000000000001e-41 or 0.46000000000000002 < eh

   1. Initial program 99.9%

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

   4. Applied rewrites 54.7%

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

   6. Applied rewrites 70.9%

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

   7. Taylor expanded in ew around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied rewrites 87.2%

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

   10. Taylor expanded in eh around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. unsub-neg N/A

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

       5. lower--.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-sin.f64 N/A

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

       12. lower-atan.f64 N/A

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

       13. lower-neg.f64 N/A

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

       14. associate-/l* N/A

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

       15. lower-*.f64 N/A

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

   12. Applied rewrites 99.8%

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

   if -2.00000000000000001e-41 < eh < 0.46000000000000002

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2 \cdot 10^{-41}:\\ \;\;\;\;\left|eh \cdot \left(\frac{ew \cdot \cos t}{eh} - \sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right|\\ \mathbf{elif}\;eh \leq 0.46:\\ \;\;\;\;\left|\frac{ew \cdot \cos t + eh \cdot \left(\sin t \cdot \left(eh \cdot \frac{\tan t}{ew}\right)\right)}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\frac{ew \cdot \cos t}{eh} - \sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\cos t\\ t_2 := \sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\\ \mathbf{if}\;ew \leq -9 \cdot 10^{-15}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{t\_2}{ew}, t\_1\right)\right|\\ \mathbf{elif}\;ew \leq 10^{-17}:\\ \;\;\;\;\left|eh \cdot \left(\frac{ew \cdot \cos t}{eh} - t\_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(-\frac{t \cdot eh}{ew}\right)}{ew}, t\_1\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (- (cos t)))
        (t_2 (* (sin t) (sin (atan (* (- eh) (/ (tan t) ew)))))))
   (if (<= ew -9e-15)
     (fabs (* ew (fma eh (/ t_2 ew) t_1)))
     (if (<= ew 1e-17)
       (fabs (* eh (- (/ (* ew (cos t)) eh) t_2)))
       (fabs
        (*
         ew
         (fma eh (/ (* (sin t) (sin (atan (- (/ (* t eh) ew))))) ew) t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -cos(t);
	double t_2 = sin(t) * sin(atan((-eh * (tan(t) / ew))));
	double tmp;
	if (ew <= -9e-15) {
		tmp = fabs((ew * fma(eh, (t_2 / ew), t_1)));
	} else if (ew <= 1e-17) {
		tmp = fabs((eh * (((ew * cos(t)) / eh) - t_2)));
	} else {
		tmp = fabs((ew * fma(eh, ((sin(t) * sin(atan(-((t * eh) / ew)))) / ew), t_1)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(-cos(t))
	t_2 = Float64(sin(t) * sin(atan(Float64(Float64(-eh) * Float64(tan(t) / ew)))))
	tmp = 0.0
	if (ew <= -9e-15)
		tmp = abs(Float64(ew * fma(eh, Float64(t_2 / ew), t_1)));
	elseif (ew <= 1e-17)
		tmp = abs(Float64(eh * Float64(Float64(Float64(ew * cos(t)) / eh) - t_2)));
	else
		tmp = abs(Float64(ew * fma(eh, Float64(Float64(sin(t) * sin(atan(Float64(-Float64(Float64(t * eh) / ew))))) / ew), t_1)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = (-N[Cos[t], $MachinePrecision])}, Block[{t$95$2 = N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[((-eh) * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -9e-15], N[Abs[N[(ew * N[(eh * N[(t$95$2 / ew), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1e-17], N[Abs[N[(eh * N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] / eh), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[(eh * N[(N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[(-N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision])], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -8.9999999999999995e-15

   1. Initial program 99.8%

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

   4. Applied rewrites 90.2%

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

   6. Applied rewrites 36.8%

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

   7. Taylor expanded in ew around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied rewrites 99.8%

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

   if -8.9999999999999995e-15 < ew < 1.00000000000000007e-17

   1. Initial program 99.9%

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

   4. Applied rewrites 57.5%

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

   6. Applied rewrites 71.3%

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

   7. Taylor expanded in ew around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied rewrites 84.5%

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

   10. Taylor expanded in eh around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. unsub-neg N/A

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

       5. lower--.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-sin.f64 N/A

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

       12. lower-atan.f64 N/A

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

       13. lower-neg.f64 N/A

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

       14. associate-/l* N/A

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

       15. lower-*.f64 N/A

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

   12. Applied rewrites 98.8%

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

   if 1.00000000000000007e-17 < ew

   1. Initial program 99.9%

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

   4. Applied rewrites 93.7%

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

   6. Applied rewrites 64.8%

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

   7. Taylor expanded in ew around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied rewrites 99.4%

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

   10. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 99.4

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

   12. Applied rewrites 99.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -9 \cdot 10^{-15}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}{ew}, -\cos t\right)\right|\\ \mathbf{elif}\;ew \leq 10^{-17}:\\ \;\;\;\;\left|eh \cdot \left(\frac{ew \cdot \cos t}{eh} - \sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(-\frac{t \cdot eh}{ew}\right)}{ew}, -\cos t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(-\frac{t \cdot eh}{ew}\right)}{ew}, -\cos t\right)\right|\\ \mathbf{if}\;ew \leq -9 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 10^{-17}:\\ \;\;\;\;\left|eh \cdot \left(\frac{ew \cdot \cos t}{eh} - \sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           ew
           (fma
            eh
            (/ (* (sin t) (sin (atan (- (/ (* t eh) ew))))) ew)
            (- (cos t)))))))
   (if (<= ew -9e-15)
     t_1
     (if (<= ew 1e-17)
       (fabs
        (*
         eh
         (-
          (/ (* ew (cos t)) eh)
          (* (sin t) (sin (atan (* (- eh) (/ (tan t) ew))))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * fma(eh, ((sin(t) * sin(atan(-((t * eh) / ew)))) / ew), -cos(t))));
	double tmp;
	if (ew <= -9e-15) {
		tmp = t_1;
	} else if (ew <= 1e-17) {
		tmp = fabs((eh * (((ew * cos(t)) / eh) - (sin(t) * sin(atan((-eh * (tan(t) / ew))))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * fma(eh, Float64(Float64(sin(t) * sin(atan(Float64(-Float64(Float64(t * eh) / ew))))) / ew), Float64(-cos(t)))))
	tmp = 0.0
	if (ew <= -9e-15)
		tmp = t_1;
	elseif (ew <= 1e-17)
		tmp = abs(Float64(eh * Float64(Float64(Float64(ew * cos(t)) / eh) - Float64(sin(t) * sin(atan(Float64(Float64(-eh) * Float64(tan(t) / ew))))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[(eh * N[(N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[(-N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision])], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + (-N[Cos[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -9e-15], t$95$1, If[LessEqual[ew, 1e-17], N[Abs[N[(eh * N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] / eh), $MachinePrecision] - N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[((-eh) * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -8.9999999999999995e-15 or 1.00000000000000007e-17 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 92.1%

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

   6. Applied rewrites 52.1%

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

   7. Taylor expanded in ew around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied rewrites 99.5%

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

   10. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 99.5

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

   12. Applied rewrites 99.5%

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

   if -8.9999999999999995e-15 < ew < 1.00000000000000007e-17

   1. Initial program 99.9%

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

   4. Applied rewrites 57.5%

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

   6. Applied rewrites 71.3%

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

   7. Taylor expanded in ew around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied rewrites 84.5%

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

   10. Taylor expanded in eh around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

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

       4. unsub-neg N/A

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

       5. lower--.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-sin.f64 N/A

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

       12. lower-atan.f64 N/A

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

       13. lower-neg.f64 N/A

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

       14. associate-/l* N/A

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

       15. lower-*.f64 N/A

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

   12. Applied rewrites 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -9 \cdot 10^{-15}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(-\frac{t \cdot eh}{ew}\right)}{ew}, -\cos t\right)\right|\\ \mathbf{elif}\;ew \leq 10^{-17}:\\ \;\;\;\;\left|eh \cdot \left(\frac{ew \cdot \cos t}{eh} - \sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(-\frac{t \cdot eh}{ew}\right)}{ew}, -\cos t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -2.6 \cdot 10^{+227}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(-\frac{t \cdot eh}{ew}\right)}{ew}, -\cos t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -2.6e+227)
   (fabs (* eh (sin t)))
   (fabs
    (*
     ew
     (fma
      eh
      (/ (* (sin t) (sin (atan (- (/ (* t eh) ew))))) ew)
      (- (cos t)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -2.6e+227) {
		tmp = fabs((eh * sin(t)));
	} else {
		tmp = fabs((ew * fma(eh, ((sin(t) * sin(atan(-((t * eh) / ew)))) / ew), -cos(t))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -2.6e+227)
		tmp = abs(Float64(eh * sin(t)));
	else
		tmp = abs(Float64(ew * fma(eh, Float64(Float64(sin(t) * sin(atan(Float64(-Float64(Float64(t * eh) / ew))))) / ew), Float64(-cos(t)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -2.6e+227], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[(eh * N[(N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[(-N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision])], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + (-N[Cos[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.59999999999999982e227

   1. Initial program 99.8%

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

   4. Applied rewrites 14.3%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 93.4

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

   7. Applied rewrites 93.4%

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

   if -2.59999999999999982e227 < eh

   1. Initial program 99.9%

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

   4. Applied rewrites 79.9%

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

   6. Applied rewrites 58.6%

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

   7. Taylor expanded in ew around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied rewrites 94.4%

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

   10. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 94.3

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

   12. Applied rewrites 94.3%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.6 \cdot 10^{+227}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(-\frac{t \cdot eh}{ew}\right)}{ew}, -\cos t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -1.3 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 4.5 \cdot 10^{-11}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= ew -1.3e+23) t_1 (if (<= ew 4.5e-11) (fabs (* eh (sin t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -1.3e+23) {
		tmp = t_1;
	} else if (ew <= 4.5e-11) {
		tmp = fabs((eh * sin(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 * cos(t)))
    if (ew <= (-1.3d+23)) then
        tmp = t_1
    else if (ew <= 4.5d-11) then
        tmp = abs((eh * sin(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.cos(t)));
	double tmp;
	if (ew <= -1.3e+23) {
		tmp = t_1;
	} else if (ew <= 4.5e-11) {
		tmp = Math.abs((eh * Math.sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if ew <= -1.3e+23:
		tmp = t_1
	elif ew <= 4.5e-11:
		tmp = math.fabs((eh * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (ew <= -1.3e+23)
		tmp = t_1;
	elseif (ew <= 4.5e-11)
		tmp = abs(Float64(eh * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * cos(t)));
	tmp = 0.0;
	if (ew <= -1.3e+23)
		tmp = t_1;
	elseif (ew <= 4.5e-11)
		tmp = abs((eh * sin(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.3e+23], t$95$1, If[LessEqual[ew, 4.5e-11], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.29999999999999996e23 or 4.5e-11 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 94.0%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 88.3

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

   7. Applied rewrites 88.3%

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

   if -1.29999999999999996e23 < ew < 4.5e-11

   1. Initial program 99.9%

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

   4. Applied rewrites 57.5%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 70.4

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

   7. Applied rewrites 70.4%

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

Alternative 7: 57.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.6 \cdot 10^{+31}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{elif}\;ew \leq 2 \cdot 10^{+132}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.6e+31)
   (fabs ew)
   (if (<= ew 2e+132) (fabs (* eh (sin t))) (fabs ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.6e+31) {
		tmp = fabs(ew);
	} else if (ew <= 2e+132) {
		tmp = fabs((eh * sin(t)));
	} else {
		tmp = fabs(ew);
	}
	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 (ew <= (-1.6d+31)) then
        tmp = abs(ew)
    else if (ew <= 2d+132) then
        tmp = abs((eh * sin(t)))
    else
        tmp = abs(ew)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.6e+31) {
		tmp = Math.abs(ew);
	} else if (ew <= 2e+132) {
		tmp = Math.abs((eh * Math.sin(t)));
	} else {
		tmp = Math.abs(ew);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1.6e+31:
		tmp = math.fabs(ew)
	elif ew <= 2e+132:
		tmp = math.fabs((eh * math.sin(t)))
	else:
		tmp = math.fabs(ew)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.6e+31)
		tmp = abs(ew);
	elseif (ew <= 2e+132)
		tmp = abs(Float64(eh * sin(t)));
	else
		tmp = abs(ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1.6e+31)
		tmp = abs(ew);
	elseif (ew <= 2e+132)
		tmp = abs((eh * sin(t)));
	else
		tmp = abs(ew);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.6e+31], N[Abs[ew], $MachinePrecision], If[LessEqual[ew, 2e+132], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[ew], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.6e31 or 1.99999999999999998e132 < ew

   1. Initial program 99.9%

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

   4. Applied rewrites 97.0%

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

   6. Applied rewrites 52.1%

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

   7. Taylor expanded in ew around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in t around 0

       \[\leadsto \left|\left(\mathsf{neg}\left(ew\right)\right) \cdot \color{blue}{-1}\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 59.3%

       \[\leadsto \left|\left(-ew\right) \cdot \color{blue}{-1}\right| \]

   if -1.6e31 < ew < 1.99999999999999998e132

   1. Initial program 99.8%

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

   4. Applied rewrites 63.3%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 63.3

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

   7. Applied rewrites 63.3%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.6 \cdot 10^{+31}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{elif}\;ew \leq 2 \cdot 10^{+132}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.5% accurate, 43.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -6.5 \cdot 10^{-112}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{elif}\;ew \leq 1.28 \cdot 10^{-194}:\\ \;\;\;\;\left|t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -6.5e-112)
   (fabs ew)
   (if (<= ew 1.28e-194) (fabs (* t eh)) (fabs ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -6.5e-112) {
		tmp = fabs(ew);
	} else if (ew <= 1.28e-194) {
		tmp = fabs((t * eh));
	} else {
		tmp = fabs(ew);
	}
	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 (ew <= (-6.5d-112)) then
        tmp = abs(ew)
    else if (ew <= 1.28d-194) then
        tmp = abs((t * eh))
    else
        tmp = abs(ew)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -6.5e-112) {
		tmp = Math.abs(ew);
	} else if (ew <= 1.28e-194) {
		tmp = Math.abs((t * eh));
	} else {
		tmp = Math.abs(ew);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -6.5e-112:
		tmp = math.fabs(ew)
	elif ew <= 1.28e-194:
		tmp = math.fabs((t * eh))
	else:
		tmp = math.fabs(ew)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -6.5e-112)
		tmp = abs(ew);
	elseif (ew <= 1.28e-194)
		tmp = abs(Float64(t * eh));
	else
		tmp = abs(ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -6.5e-112)
		tmp = abs(ew);
	elseif (ew <= 1.28e-194)
		tmp = abs((t * eh));
	else
		tmp = abs(ew);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -6.5e-112], N[Abs[ew], $MachinePrecision], If[LessEqual[ew, 1.28e-194], N[Abs[N[(t * eh), $MachinePrecision]], $MachinePrecision], N[Abs[ew], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -6.49999999999999956e-112 or 1.2800000000000001e-194 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 86.7%

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

   6. Applied rewrites 55.8%

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

   7. Taylor expanded in ew around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. rem-square-sqrt N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied rewrites 97.8%

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

   10. Taylor expanded in t around 0

       \[\leadsto \left|\left(\mathsf{neg}\left(ew\right)\right) \cdot \color{blue}{-1}\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 46.5%

       \[\leadsto \left|\left(-ew\right) \cdot \color{blue}{-1}\right| \]

   if -6.49999999999999956e-112 < ew < 1.2800000000000001e-194

   1. Initial program 99.9%

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 81.6

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

   7. Applied rewrites 81.6%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 46.4

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

   10. Applied rewrites 46.4%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -6.5 \cdot 10^{-112}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{elif}\;ew \leq 1.28 \cdot 10^{-194}:\\ \;\;\;\;\left|t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 18.8% accurate, 107.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 76.0%

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

5. Taylor expanded in ew around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 42.5

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

7. Applied rewrites 42.5%

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

8. Taylor expanded in t around 0

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

   1. lower-*.f64 21.0

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

10. Applied rewrites 21.0%

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

11. Final simplification 21.0%

    \[\leadsto \left|t \cdot eh\right| \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))