Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 16.9s

Alternatives: 12

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: 98.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 41.4%

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

5. Taylor expanded in eh around 0

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

   1. mul-1-neg N/A

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

   2. associate-*r* N/A

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

   3. distribute-lft-neg-in N/A

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

   4. mul-1-neg N/A

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

   5. lower-fma.f64 N/A

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

7. Applied rewrites 98.5%

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

Alternative 3: 98.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 41.4%

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

5. Taylor expanded in eh around 0

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

   1. mul-1-neg N/A

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

   2. associate-*r* N/A

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

   3. distribute-lft-neg-in N/A

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

   4. mul-1-neg N/A

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

   5. lower-fma.f64 N/A

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 98.5%

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

11. Final simplification 98.5%

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

Alternative 4: 78.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot eh}{ew}\\ t_2 := eh \cdot \sin t\\ \mathbf{if}\;eh \leq -1.05 \cdot 10^{+117}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \frac{ew \cdot \mathsf{fma}\left(0.5, \cos \left(t + t\right), 0.5\right)}{t\_2}, \sin t\right)\right|\\ \mathbf{elif}\;eh \leq 6.8 \cdot 10^{+35}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{t \cdot eh}{\left(-ew\right) \cdot \sqrt{\mathsf{fma}\left(t\_1, t\_1, 1\right)}}, -t\_2, ew \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \frac{ew \cdot 1}{t\_2}, \sin t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (* t eh) ew)) (t_2 (* eh (sin t))))
   (if (<= eh -1.05e+117)
     (fabs
      (*
       eh
       (fma
        0.5
        (* (/ ew eh) (/ (* ew (fma 0.5 (cos (+ t t)) 0.5)) t_2))
        (sin t))))
     (if (<= eh 6.8e+35)
       (fabs
        (fma
         (/ (* t eh) (* (- ew) (sqrt (fma t_1 t_1 1.0))))
         (- t_2)
         (* ew (cos t))))
       (fabs (* eh (fma 0.5 (* (/ ew eh) (/ (* ew 1.0) t_2)) (sin t))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (t * eh) / ew;
	double t_2 = eh * sin(t);
	double tmp;
	if (eh <= -1.05e+117) {
		tmp = fabs((eh * fma(0.5, ((ew / eh) * ((ew * fma(0.5, cos((t + t)), 0.5)) / t_2)), sin(t))));
	} else if (eh <= 6.8e+35) {
		tmp = fabs(fma(((t * eh) / (-ew * sqrt(fma(t_1, t_1, 1.0)))), -t_2, (ew * cos(t))));
	} else {
		tmp = fabs((eh * fma(0.5, ((ew / eh) * ((ew * 1.0) / t_2)), sin(t))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(t * eh) / ew)
	t_2 = Float64(eh * sin(t))
	tmp = 0.0
	if (eh <= -1.05e+117)
		tmp = abs(Float64(eh * fma(0.5, Float64(Float64(ew / eh) * Float64(Float64(ew * fma(0.5, cos(Float64(t + t)), 0.5)) / t_2)), sin(t))));
	elseif (eh <= 6.8e+35)
		tmp = abs(fma(Float64(Float64(t * eh) / Float64(Float64(-ew) * sqrt(fma(t_1, t_1, 1.0)))), Float64(-t_2), Float64(ew * cos(t))));
	else
		tmp = abs(Float64(eh * fma(0.5, Float64(Float64(ew / eh) * Float64(Float64(ew * 1.0) / t_2)), sin(t))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -1.05e+117], N[Abs[N[(eh * N[(0.5 * N[(N[(ew / eh), $MachinePrecision] * N[(N[(ew * N[(0.5 * N[Cos[N[(t + t), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] + N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 6.8e+35], N[Abs[N[(N[(N[(t * eh), $MachinePrecision] / N[((-ew) * N[Sqrt[N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-t$95$2) + N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[(0.5 * N[(N[(ew / eh), $MachinePrecision] * N[(N[(ew * 1.0), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] + N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -1.0500000000000001e117

   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
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 44.2%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 76.0%

      \[\leadsto \left|\color{blue}{eh \cdot \mathsf{fma}\left(0.5, \frac{\left(ew \cdot ew\right) \cdot {\cos t}^{2}}{\left(eh \cdot eh\right) \cdot \sin t}, \sin t\right)}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 79.8%

      \[\leadsto \left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \color{blue}{\frac{ew \cdot \mathsf{fma}\left(0.5, \cos \left(t + t\right), 0.5\right)}{eh \cdot \sin t}}, \sin t\right)\right| \]

   if -1.0500000000000001e117 < eh < 6.8000000000000002e35

   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 53.1%

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

   5. Taylor expanded in eh around 0

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 98.5%

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

   12. Applied rewrites 87.9%

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

   if 6.8000000000000002e35 < eh

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 43.3%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 83.1%

      \[\leadsto \left|\color{blue}{eh \cdot \mathsf{fma}\left(0.5, \frac{\left(ew \cdot ew\right) \cdot {\cos t}^{2}}{\left(eh \cdot eh\right) \cdot \sin t}, \sin t\right)}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 83.5%

      \[\leadsto \left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \color{blue}{\frac{ew \cdot \mathsf{fma}\left(0.5, \cos \left(t + t\right), 0.5\right)}{eh \cdot \sin t}}, \sin t\right)\right| \]

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 85.4%

       \[\leadsto \left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \frac{ew \cdot 1}{eh \cdot \sin t}, \sin t\right)\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.05 \cdot 10^{+117}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \frac{ew \cdot \mathsf{fma}\left(0.5, \cos \left(t + t\right), 0.5\right)}{eh \cdot \sin t}, \sin t\right)\right|\\ \mathbf{elif}\;eh \leq 6.8 \cdot 10^{+35}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{t \cdot eh}{\left(-ew\right) \cdot \sqrt{\mathsf{fma}\left(\frac{t \cdot eh}{ew}, \frac{t \cdot eh}{ew}, 1\right)}}, -eh \cdot \sin t, ew \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \frac{ew \cdot 1}{eh \cdot \sin t}, \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ t_2 := \left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \frac{ew \cdot 1}{t\_1}, \sin t\right)\right|\\ t_3 := \frac{t \cdot eh}{ew}\\ \mathbf{if}\;eh \leq -1 \cdot 10^{+117}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 6.8 \cdot 10^{+35}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{t \cdot eh}{\left(-ew\right) \cdot \sqrt{\mathsf{fma}\left(t\_3, t\_3, 1\right)}}, -t\_1, ew \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t)))
        (t_2 (fabs (* eh (fma 0.5 (* (/ ew eh) (/ (* ew 1.0) t_1)) (sin t)))))
        (t_3 (/ (* t eh) ew)))
   (if (<= eh -1e+117)
     t_2
     (if (<= eh 6.8e+35)
       (fabs
        (fma
         (/ (* t eh) (* (- ew) (sqrt (fma t_3 t_3 1.0))))
         (- t_1)
         (* ew (cos t))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double t_2 = fabs((eh * fma(0.5, ((ew / eh) * ((ew * 1.0) / t_1)), sin(t))));
	double t_3 = (t * eh) / ew;
	double tmp;
	if (eh <= -1e+117) {
		tmp = t_2;
	} else if (eh <= 6.8e+35) {
		tmp = fabs(fma(((t * eh) / (-ew * sqrt(fma(t_3, t_3, 1.0)))), -t_1, (ew * cos(t))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	t_2 = abs(Float64(eh * fma(0.5, Float64(Float64(ew / eh) * Float64(Float64(ew * 1.0) / t_1)), sin(t))))
	t_3 = Float64(Float64(t * eh) / ew)
	tmp = 0.0
	if (eh <= -1e+117)
		tmp = t_2;
	elseif (eh <= 6.8e+35)
		tmp = abs(fma(Float64(Float64(t * eh) / Float64(Float64(-ew) * sqrt(fma(t_3, t_3, 1.0)))), Float64(-t_1), Float64(ew * cos(t))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(eh * N[(0.5 * N[(N[(ew / eh), $MachinePrecision] * N[(N[(ew * 1.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]}, If[LessEqual[eh, -1e+117], t$95$2, If[LessEqual[eh, 6.8e+35], N[Abs[N[(N[(N[(t * eh), $MachinePrecision] / N[((-ew) * N[Sqrt[N[(t$95$3 * t$95$3 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-t$95$1) + N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.00000000000000005e117 or 6.8000000000000002e35 < eh

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 43.7%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 79.9%

      \[\leadsto \left|\color{blue}{eh \cdot \mathsf{fma}\left(0.5, \frac{\left(ew \cdot ew\right) \cdot {\cos t}^{2}}{\left(eh \cdot eh\right) \cdot \sin t}, \sin t\right)}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 81.8%

      \[\leadsto \left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \color{blue}{\frac{ew \cdot \mathsf{fma}\left(0.5, \cos \left(t + t\right), 0.5\right)}{eh \cdot \sin t}}, \sin t\right)\right| \]

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 82.9%

       \[\leadsto \left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \frac{ew \cdot 1}{eh \cdot \sin t}, \sin t\right)\right| \]

   if -1.00000000000000005e117 < eh < 6.8000000000000002e35

   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 53.1%

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

   5. Taylor expanded in eh around 0

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 98.5%

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

   12. Applied rewrites 87.9%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1 \cdot 10^{+117}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \frac{ew \cdot 1}{eh \cdot \sin t}, \sin t\right)\right|\\ \mathbf{elif}\;eh \leq 6.8 \cdot 10^{+35}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{t \cdot eh}{\left(-ew\right) \cdot \sqrt{\mathsf{fma}\left(\frac{t \cdot eh}{ew}, \frac{t \cdot eh}{ew}, 1\right)}}, -eh \cdot \sin t, ew \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \frac{ew \cdot 1}{eh \cdot \sin t}, \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \frac{ew \cdot 1}{eh \cdot \sin t}, \sin t\right)\right|\\ \mathbf{if}\;eh \leq -2.2 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 6.8 \cdot 10^{+35}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           eh
           (fma 0.5 (* (/ ew eh) (/ (* ew 1.0) (* eh (sin t)))) (sin t))))))
   (if (<= eh -2.2e+91) t_1 (if (<= eh 6.8e+35) (fabs (* ew (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * fma(0.5, ((ew / eh) * ((ew * 1.0) / (eh * sin(t)))), sin(t))));
	double tmp;
	if (eh <= -2.2e+91) {
		tmp = t_1;
	} else if (eh <= 6.8e+35) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * fma(0.5, Float64(Float64(ew / eh) * Float64(Float64(ew * 1.0) / Float64(eh * sin(t)))), sin(t))))
	tmp = 0.0
	if (eh <= -2.2e+91)
		tmp = t_1;
	elseif (eh <= 6.8e+35)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[(0.5 * N[(N[(ew / eh), $MachinePrecision] * N[(N[(ew * 1.0), $MachinePrecision] / N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.2e+91], t$95$1, If[LessEqual[eh, 6.8e+35], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.19999999999999999e91 or 6.8000000000000002e35 < 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
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 46.0%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 79.8%

      \[\leadsto \left|\color{blue}{eh \cdot \mathsf{fma}\left(0.5, \frac{\left(ew \cdot ew\right) \cdot {\cos t}^{2}}{\left(eh \cdot eh\right) \cdot \sin t}, \sin t\right)}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 81.6%

      \[\leadsto \left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \color{blue}{\frac{ew \cdot \mathsf{fma}\left(0.5, \cos \left(t + t\right), 0.5\right)}{eh \cdot \sin t}}, \sin t\right)\right| \]

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 82.6%

       \[\leadsto \left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \frac{ew \cdot 1}{eh \cdot \sin t}, \sin t\right)\right| \]

   if -2.19999999999999999e91 < eh < 6.8000000000000002e35

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 94.3%

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

   5. Taylor expanded in ew around 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 77.2

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

   7. Applied rewrites 77.2%

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

Alternative 7: 74.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \frac{ew}{t \cdot eh}, \sin t\right)\right|\\ \mathbf{if}\;eh \leq -2.2 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 6.8 \cdot 10^{+35}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (fma 0.5 (* (/ ew eh) (/ ew (* t eh))) (sin t))))))
   (if (<= eh -2.2e+91) t_1 (if (<= eh 6.8e+35) (fabs (* ew (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * fma(0.5, ((ew / eh) * (ew / (t * eh))), sin(t))));
	double tmp;
	if (eh <= -2.2e+91) {
		tmp = t_1;
	} else if (eh <= 6.8e+35) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * fma(0.5, Float64(Float64(ew / eh) * Float64(ew / Float64(t * eh))), sin(t))))
	tmp = 0.0
	if (eh <= -2.2e+91)
		tmp = t_1;
	elseif (eh <= 6.8e+35)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[(0.5 * N[(N[(ew / eh), $MachinePrecision] * N[(ew / N[(t * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.2e+91], t$95$1, If[LessEqual[eh, 6.8e+35], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.19999999999999999e91 or 6.8000000000000002e35 < 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
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 46.0%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 79.8%

      \[\leadsto \left|\color{blue}{eh \cdot \mathsf{fma}\left(0.5, \frac{\left(ew \cdot ew\right) \cdot {\cos t}^{2}}{\left(eh \cdot eh\right) \cdot \sin t}, \sin t\right)}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 81.6%

      \[\leadsto \left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \color{blue}{\frac{ew \cdot \mathsf{fma}\left(0.5, \cos \left(t + t\right), 0.5\right)}{eh \cdot \sin t}}, \sin t\right)\right| \]

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 82.5%

       \[\leadsto \left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \frac{ew}{\color{blue}{eh \cdot t}}, \sin t\right)\right| \]

   if -2.19999999999999999e91 < eh < 6.8000000000000002e35

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 94.3%

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

   5. Taylor expanded in ew around 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 77.2

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

   7. Applied rewrites 77.2%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.2 \cdot 10^{+91}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \frac{ew}{t \cdot eh}, \sin t\right)\right|\\ \mathbf{elif}\;eh \leq 6.8 \cdot 10^{+35}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(0.5, \frac{ew}{eh} \cdot \frac{ew}{t \cdot eh}, \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;eh \leq -2.2 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 6.8 \cdot 10^{+35}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= eh -2.2e+91) t_1 (if (<= eh 6.8e+35) (fabs (* ew (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (eh <= -2.2e+91) {
		tmp = t_1;
	} else if (eh <= 6.8e+35) {
		tmp = fabs((ew * cos(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((eh * sin(t)))
    if (eh <= (-2.2d+91)) then
        tmp = t_1
    else if (eh <= 6.8d+35) then
        tmp = abs((ew * cos(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((eh * Math.sin(t)));
	double tmp;
	if (eh <= -2.2e+91) {
		tmp = t_1;
	} else if (eh <= 6.8e+35) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	tmp = 0
	if eh <= -2.2e+91:
		tmp = t_1
	elif eh <= 6.8e+35:
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (eh <= -2.2e+91)
		tmp = t_1;
	elseif (eh <= 6.8e+35)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	tmp = 0.0;
	if (eh <= -2.2e+91)
		tmp = t_1;
	elseif (eh <= 6.8e+35)
		tmp = abs((ew * cos(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.2e+91], t$95$1, If[LessEqual[eh, 6.8e+35], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.19999999999999999e91 or 6.8000000000000002e35 < 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
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 46.0%

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

   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 82.1

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

   7. Applied rewrites 82.1%

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

   if -2.19999999999999999e91 < eh < 6.8000000000000002e35

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 94.3%

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

   5. Taylor expanded in ew around 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 77.2

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

   7. Applied rewrites 77.2%

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

Alternative 9: 59.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{-49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{-160}:\\ \;\;\;\;\left|\mathsf{fma}\left(t \cdot t, ew \cdot -0.5, ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= t -4.2e-49)
     t_1
     (if (<= t 1.16e-160) (fabs (fma (* t t) (* ew -0.5) ew)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (t <= -4.2e-49) {
		tmp = t_1;
	} else if (t <= 1.16e-160) {
		tmp = fabs(fma((t * t), (ew * -0.5), ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (t <= -4.2e-49)
		tmp = t_1;
	elseif (t <= 1.16e-160)
		tmp = abs(fma(Float64(t * t), Float64(ew * -0.5), ew));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -4.2e-49], t$95$1, If[LessEqual[t, 1.16e-160], N[Abs[N[(N[(t * t), $MachinePrecision] * N[(ew * -0.5), $MachinePrecision] + ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.1999999999999998e-49 or 1.16e-160 < t

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 67.9%

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

   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 60.4

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

   7. Applied rewrites 60.4%

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

   if -4.1999999999999998e-49 < t < 1.16e-160

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 75.6%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(ew, -0.5, \frac{0.5 \cdot \left(eh \cdot eh\right)}{ew}\right), ew\right)}\right| \]

   8. Taylor expanded in ew around inf

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

   10. Applied rewrites 80.9%

       \[\leadsto \left|\mathsf{fma}\left(t \cdot t, ew \cdot \color{blue}{-0.5}, ew\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 36.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq 5.2 \cdot 10^{+31}:\\ \;\;\;\;\left|\mathsf{fma}\left(t \cdot \left(ew \cdot -0.5\right), t, ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(eh \cdot \left(t \cdot t\right), \mathsf{fma}\left(0.5, \frac{ew \cdot ew}{eh \cdot eh} \cdot -0.8333333333333334, 1\right), \frac{0.5 \cdot \left(ew \cdot ew\right)}{eh}\right)}{t}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh 5.2e+31)
   (fabs (fma (* t (* ew -0.5)) t ew))
   (fabs
    (/
     (fma
      (* eh (* t t))
      (fma 0.5 (* (/ (* ew ew) (* eh eh)) -0.8333333333333334) 1.0)
      (/ (* 0.5 (* ew ew)) eh))
     t))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 5.2e+31) {
		tmp = fabs(fma((t * (ew * -0.5)), t, ew));
	} else {
		tmp = fabs((fma((eh * (t * t)), fma(0.5, (((ew * ew) / (eh * eh)) * -0.8333333333333334), 1.0), ((0.5 * (ew * ew)) / eh)) / t));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= 5.2e+31)
		tmp = abs(fma(Float64(t * Float64(ew * -0.5)), t, ew));
	else
		tmp = abs(Float64(fma(Float64(eh * Float64(t * t)), fma(0.5, Float64(Float64(Float64(ew * ew) / Float64(eh * eh)) * -0.8333333333333334), 1.0), Float64(Float64(0.5 * Float64(ew * ew)) / eh)) / t));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, 5.2e+31], N[Abs[N[(N[(t * N[(ew * -0.5), $MachinePrecision]), $MachinePrecision] * t + ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] * N[(0.5 * N[(N[(N[(ew * ew), $MachinePrecision] / N[(eh * eh), $MachinePrecision]), $MachinePrecision] * -0.8333333333333334), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(0.5 * N[(ew * ew), $MachinePrecision]), $MachinePrecision] / eh), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 5.2e31

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 84.2%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   7. Applied rewrites 39.4%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(ew, -0.5, \frac{0.5 \cdot \left(eh \cdot eh\right)}{ew}\right), ew\right)}\right| \]

   8. Taylor expanded in ew around inf

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

   10. Applied rewrites 41.2%

       \[\leadsto \left|\mathsf{fma}\left(t \cdot t, ew \cdot \color{blue}{-0.5}, ew\right)\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 41.3%

       \[\leadsto \left|\mathsf{fma}\left(t \cdot \left(ew \cdot -0.5\right), \color{blue}{t}, ew\right)\right| \]

   if 5.2e31 < eh

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 46.4%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 80.5%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 27.9%

       \[\leadsto \left|\frac{\mathsf{fma}\left(eh \cdot \left(t \cdot t\right), \mathsf{fma}\left(0.5, \frac{ew \cdot ew}{eh \cdot eh} \cdot -0.8333333333333334, 1\right), \frac{0.5 \cdot \left(ew \cdot ew\right)}{eh}\right)}{\color{blue}{t}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 38.8% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(t \cdot \left(ew \cdot -0.5\right), t, ew\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (fma (* t (* ew -0.5)) t ew)))

C

double code(double eh, double ew, double t) {
	return fabs(fma((t * (ew * -0.5)), t, ew));
}

Julia

function code(eh, ew, t)
	return abs(fma(Float64(t * Float64(ew * -0.5)), t, ew))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(t * N[(ew * -0.5), $MachinePrecision]), $MachinePrecision] * t + ew), $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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-atan.f64 N/A

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

   5. cos-atan N/A

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

   6. un-div-inv N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

4. Applied rewrites 76.2%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f64 N/A

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

7. Applied rewrites 33.3%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(ew, -0.5, \frac{0.5 \cdot \left(eh \cdot eh\right)}{ew}\right), ew\right)}\right| \]

8. Taylor expanded in ew around inf

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

10. Applied rewrites 35.9%

    \[\leadsto \left|\mathsf{fma}\left(t \cdot t, ew \cdot \color{blue}{-0.5}, ew\right)\right| \]
11. Step-by-step derivation

12. Applied rewrites 35.9%

    \[\leadsto \left|\mathsf{fma}\left(t \cdot \left(ew \cdot -0.5\right), \color{blue}{t}, ew\right)\right| \]
13. Add Preprocessing

Alternative 12: 38.7% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(t \cdot t, ew \cdot -0.5, ew\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (fma (* t t) (* ew -0.5) ew)))

C

double code(double eh, double ew, double t) {
	return fabs(fma((t * t), (ew * -0.5), ew));
}

Julia

function code(eh, ew, t)
	return abs(fma(Float64(t * t), Float64(ew * -0.5), ew))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(t * t), $MachinePrecision] * N[(ew * -0.5), $MachinePrecision] + ew), $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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-atan.f64 N/A

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

   5. cos-atan N/A

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

   6. un-div-inv N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

4. Applied rewrites 76.2%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f64 N/A

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

7. Applied rewrites 33.3%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(ew, -0.5, \frac{0.5 \cdot \left(eh \cdot eh\right)}{ew}\right), ew\right)}\right| \]

8. Taylor expanded in ew around inf

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

10. Applied rewrites 35.9%

    \[\leadsto \left|\mathsf{fma}\left(t \cdot t, ew \cdot \color{blue}{-0.5}, ew\right)\right| \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024233 
(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)))))))