Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 17.6s

Alternatives: 15

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

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = abs((((eh * sin(t)) * sin(t_1)) - ((ew * cos(t)) * cos(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[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[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%

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

Alternative 2: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ t_2 := \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -2 \cdot 10^{-263}:\\ \;\;\;\;\left|\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{\tan t \cdot eh}{-ew}\right), \frac{\sin t}{ew} \cdot \left(-eh\right), \cos \tan^{-1} \left(\mathsf{fma}\left(\frac{eh}{ew} \cdot 0.3333333333333333, t \cdot t, \frac{eh}{ew}\right) \cdot \left(-t\right)\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin t\_2 \cdot \sin t, eh, \left(\cos t \cdot ew\right) \cdot \cos t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew))))
        (t_2 (atan (* (/ (tan t) ew) eh))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -2e-263)
     (fabs
      (*
       (fma
        (sin (atan (/ (* (tan t) eh) (- ew))))
        (* (/ (sin t) ew) (- eh))
        (*
         (cos
          (atan
           (* (fma (* (/ eh ew) 0.3333333333333333) (* t t) (/ eh ew)) (- t))))
         (cos t)))
       ew))
     (fma (* (sin t_2) (sin t)) eh (* (* (cos t) ew) (cos t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double t_2 = atan(((tan(t) / ew) * eh));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -2e-263) {
		tmp = fabs((fma(sin(atan(((tan(t) * eh) / -ew))), ((sin(t) / ew) * -eh), (cos(atan((fma(((eh / ew) * 0.3333333333333333), (t * t), (eh / ew)) * -t))) * cos(t))) * ew));
	} else {
		tmp = fma((sin(t_2) * sin(t)), eh, ((cos(t) * ew) * cos(t_2)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	t_2 = atan(Float64(Float64(tan(t) / ew) * eh))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -2e-263)
		tmp = abs(Float64(fma(sin(atan(Float64(Float64(tan(t) * eh) / Float64(-ew)))), Float64(Float64(sin(t) / ew) * Float64(-eh)), Float64(cos(atan(Float64(fma(Float64(Float64(eh / ew) * 0.3333333333333333), Float64(t * t), Float64(eh / ew)) * Float64(-t)))) * cos(t))) * ew));
	else
		tmp = fma(Float64(sin(t_2) * sin(t)), eh, Float64(Float64(cos(t) * ew) * cos(t_2)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -2e-263], N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * (-eh)), $MachinePrecision] + N[(N[Cos[N[ArcTan[N[(N[(N[(N[(eh / ew), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] * (-t)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sin[t$95$2], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * eh + N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -2e-263

   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. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 82.8%

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

   7. Applied rewrites 88.1%

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

   9. Applied rewrites 88.1%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 70.5%

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

   if -2e-263 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   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-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 99.7

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{-263}:\\ \;\;\;\;\left|\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{\tan t \cdot eh}{-ew}\right), \frac{\sin t}{ew} \cdot \left(-eh\right), \cos \tan^{-1} \left(\mathsf{fma}\left(\frac{eh}{ew} \cdot 0.3333333333333333, t \cdot t, \frac{eh}{ew}\right) \cdot \left(-t\right)\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot \sin t, eh, \left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -2 \cdot 10^{-263}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{-eh}{\mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.041666666666666664 - 0.5, t \cdot t, 1\right)}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos t \cdot ew + \left(\sin t \cdot eh\right) \cdot \left(\frac{eh}{ew} \cdot \tan t\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -2e-263)
     (fabs
      (*
       (cos
        (atan
         (*
          (/ (sin t) ew)
          (/
           (- eh)
           (fma (- (* (* t t) 0.041666666666666664) 0.5) (* t t) 1.0)))))
       ew))
     (/ (+ (* (cos t) ew) (* (* (sin t) eh) (* (/ eh ew) (tan t)))) 1.0))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -2e-263) {
		tmp = fabs((cos(atan(((sin(t) / ew) * (-eh / fma((((t * t) * 0.041666666666666664) - 0.5), (t * t), 1.0))))) * ew));
	} else {
		tmp = ((cos(t) * ew) + ((sin(t) * eh) * ((eh / ew) * tan(t)))) / 1.0;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -2e-263)
		tmp = abs(Float64(cos(atan(Float64(Float64(sin(t) / ew) * Float64(Float64(-eh) / fma(Float64(Float64(Float64(t * t) * 0.041666666666666664) - 0.5), Float64(t * t), 1.0))))) * ew));
	else
		tmp = Float64(Float64(Float64(cos(t) * ew) + Float64(Float64(sin(t) * eh) * Float64(Float64(eh / ew) * tan(t)))) / 1.0);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -2e-263], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * N[((-eh) / N[(N[(N[(N[(t * t), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -2e-263

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{1 + {t}^{2} \cdot \left(\frac{1}{24} \cdot {t}^{2} - \frac{1}{2}\right)}\right) \cdot ew\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 41.5%

      \[\leadsto \left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.041666666666666664 - 0.5, t \cdot t, 1\right)}\right) \cdot ew\right| \]

   if -2e-263 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 60.9%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{-263}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{-eh}{\mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.041666666666666664 - 0.5, t \cdot t, 1\right)}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos t \cdot ew + \left(\sin t \cdot eh\right) \cdot \left(\frac{eh}{ew} \cdot \tan t\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -2 \cdot 10^{-263}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{-eh}{\mathsf{fma}\left(t \cdot t, -0.5, 1\right)}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos t \cdot ew + \left(\sin t \cdot eh\right) \cdot \left(\frac{eh}{ew} \cdot \tan t\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -2e-263)
     (fabs
      (* (cos (atan (* (/ (sin t) ew) (/ (- eh) (fma (* t t) -0.5 1.0))))) ew))
     (/ (+ (* (cos t) ew) (* (* (sin t) eh) (* (/ eh ew) (tan t)))) 1.0))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -2e-263) {
		tmp = fabs((cos(atan(((sin(t) / ew) * (-eh / fma((t * t), -0.5, 1.0))))) * ew));
	} else {
		tmp = ((cos(t) * ew) + ((sin(t) * eh) * ((eh / ew) * tan(t)))) / 1.0;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -2e-263)
		tmp = abs(Float64(cos(atan(Float64(Float64(sin(t) / ew) * Float64(Float64(-eh) / fma(Float64(t * t), -0.5, 1.0))))) * ew));
	else
		tmp = Float64(Float64(Float64(cos(t) * ew) + Float64(Float64(sin(t) * eh) * Float64(Float64(eh / ew) * tan(t)))) / 1.0);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -2e-263], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * N[((-eh) / N[(N[(t * t), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -2e-263

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 41.5%

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

   if -2e-263 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 60.9%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{-263}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{-eh}{\mathsf{fma}\left(t \cdot t, -0.5, 1\right)}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos t \cdot ew + \left(\sin t \cdot eh\right) \cdot \left(\frac{eh}{ew} \cdot \tan t\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -2 \cdot 10^{-263}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{-eh}{\mathsf{fma}\left(t \cdot t, -0.5, 1\right)}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -2e-263)
     (fabs
      (* (cos (atan (* (/ (sin t) ew) (/ (- eh) (fma (* t t) -0.5 1.0))))) ew))
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -2e-263) {
		tmp = fabs((cos(atan(((sin(t) / ew) * (-eh / fma((t * t), -0.5, 1.0))))) * ew));
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -2e-263)
		tmp = abs(Float64(cos(atan(Float64(Float64(sin(t) / ew) * Float64(Float64(-eh) / fma(Float64(t * t), -0.5, 1.0))))) * ew));
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -2e-263], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * N[((-eh) / N[(N[(t * t), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -2e-263

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 41.5%

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

   if -2e-263 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 26.8

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

   7. Applied rewrites 26.8%

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

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 60.6

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

   10. Applied rewrites 60.6%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{-263}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{-eh}{\mathsf{fma}\left(t \cdot t, -0.5, 1\right)}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -2 \cdot 10^{-263}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -2e-263)
     (fabs (* (cos (atan (* (/ (tan t) ew) eh))) ew))
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -2e-263) {
		tmp = fabs((cos(atan(((tan(t) / ew) * eh))) * ew));
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan(((eh * tan(t)) / -ew))
    if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= (-2d-263)) then
        tmp = abs((cos(atan(((tan(t) / ew) * eh))) * ew))
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if ((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))) <= -2e-263) {
		tmp = Math.abs((Math.cos(Math.atan(((Math.tan(t) / ew) * eh))) * ew));
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if (((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))) <= -2e-263:
		tmp = math.fabs((math.cos(math.atan(((math.tan(t) / ew) * eh))) * ew))
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -2e-263)
		tmp = abs(Float64(cos(atan(Float64(Float64(tan(t) / ew) * eh))) * ew));
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -2e-263)
		tmp = abs((cos(atan(((tan(t) / ew) * eh))) * ew));
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -2e-263], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -2e-263

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 41.5%

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

   7. Applied rewrites 41.5%

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

   if -2e-263 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 26.8

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

   7. Applied rewrites 26.8%

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

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 60.6

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

   10. Applied rewrites 60.6%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{-263}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\mathsf{PI}\left(\right)}\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -2 \cdot 10^{-263}:\\ \;\;\;\;\left|\sin \left(\mathsf{fma}\left(t\_1, \frac{t\_1}{2}, -\tan^{-1} \left(\frac{eh \cdot t}{ew}\right)\right)\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sqrt (PI))) (t_2 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_2)) (* (* eh (sin t)) (sin t_2)))
        -2e-263)
     (fabs (* (sin (fma t_1 (/ t_1 2.0) (- (atan (/ (* eh t) ew))))) ew))
     (* (cos t) ew))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -2e-263

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 41.5%

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

   7. Applied rewrites 41.2%

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

   9. Applied rewrites 41.5%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 40.7%

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

   if -2e-263 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 26.8

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

   7. Applied rewrites 26.8%

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

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 60.6

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

   10. Applied rewrites 60.6%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{-263}:\\ \;\;\;\;\left|\sin \left(\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \frac{\sqrt{\mathsf{PI}\left(\right)}}{2}, -\tan^{-1} \left(\frac{eh \cdot t}{ew}\right)\right)\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -2 \cdot 10^{-263}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{-eh}{ew} \cdot t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -2e-263)
     (fabs (* (cos (atan (* (/ (- eh) ew) t))) ew))
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -2e-263) {
		tmp = fabs((cos(atan(((-eh / ew) * t))) * ew));
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan(((eh * tan(t)) / -ew))
    if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= (-2d-263)) then
        tmp = abs((cos(atan(((-eh / ew) * t))) * ew))
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if ((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))) <= -2e-263) {
		tmp = Math.abs((Math.cos(Math.atan(((-eh / ew) * t))) * ew));
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if (((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))) <= -2e-263:
		tmp = math.fabs((math.cos(math.atan(((-eh / ew) * t))) * ew))
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -2e-263)
		tmp = abs(Float64(cos(atan(Float64(Float64(Float64(-eh) / ew) * t))) * ew));
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -2e-263)
		tmp = abs((cos(atan(((-eh / ew) * t))) * ew));
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -2e-263], N[Abs[N[(N[Cos[N[ArcTan[N[(N[((-eh) / ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -2e-263

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 39.0%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 40.7%

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

   if -2e-263 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. distribute-rgt-out N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 26.8

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

   7. Applied rewrites 26.8%

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

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 60.6

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

   10. Applied rewrites 60.6%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{-263}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{-eh}{ew} \cdot t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin t\\ t_2 := \left|\cos \tan^{-1} \left(\frac{t\_1}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ t_3 := \left|\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{\tan t \cdot eh}{-ew}\right), \frac{\sin t}{ew} \cdot \left(-eh\right), \cos \tan^{-1} \left(\mathsf{fma}\left(\frac{eh}{ew} \cdot 0.3333333333333333, t \cdot t, \frac{eh}{ew}\right) \cdot \left(-t\right)\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{if}\;ew \leq -2.8 \cdot 10^{+81}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq -4 \cdot 10^{-193}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;ew \leq 2.15 \cdot 10^{-178}:\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right) \cdot t\_1\right) \cdot eh\right|\\ \mathbf{elif}\;ew \leq 3.1 \cdot 10^{+37}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (- (sin t)))
        (t_2
         (fabs (* (cos (atan (* (/ t_1 ew) (/ eh (cos t))))) (* (cos t) ew))))
        (t_3
         (fabs
          (*
           (fma
            (sin (atan (/ (* (tan t) eh) (- ew))))
            (* (/ (sin t) ew) (- eh))
            (*
             (cos
              (atan
               (*
                (fma (* (/ eh ew) 0.3333333333333333) (* t t) (/ eh ew))
                (- t))))
             (cos t)))
           ew))))
   (if (<= ew -2.8e+81)
     t_2
     (if (<= ew -4e-193)
       t_3
       (if (<= ew 2.15e-178)
         (fabs (* (* (tanh (asinh (* (/ (tan t) ew) (- eh)))) t_1) eh))
         (if (<= ew 3.1e+37) t_3 t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -sin(t);
	double t_2 = fabs((cos(atan(((t_1 / ew) * (eh / cos(t))))) * (cos(t) * ew)));
	double t_3 = fabs((fma(sin(atan(((tan(t) * eh) / -ew))), ((sin(t) / ew) * -eh), (cos(atan((fma(((eh / ew) * 0.3333333333333333), (t * t), (eh / ew)) * -t))) * cos(t))) * ew));
	double tmp;
	if (ew <= -2.8e+81) {
		tmp = t_2;
	} else if (ew <= -4e-193) {
		tmp = t_3;
	} else if (ew <= 2.15e-178) {
		tmp = fabs(((tanh(asinh(((tan(t) / ew) * -eh))) * t_1) * eh));
	} else if (ew <= 3.1e+37) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(-sin(t))
	t_2 = abs(Float64(cos(atan(Float64(Float64(t_1 / ew) * Float64(eh / cos(t))))) * Float64(cos(t) * ew)))
	t_3 = abs(Float64(fma(sin(atan(Float64(Float64(tan(t) * eh) / Float64(-ew)))), Float64(Float64(sin(t) / ew) * Float64(-eh)), Float64(cos(atan(Float64(fma(Float64(Float64(eh / ew) * 0.3333333333333333), Float64(t * t), Float64(eh / ew)) * Float64(-t)))) * cos(t))) * ew))
	tmp = 0.0
	if (ew <= -2.8e+81)
		tmp = t_2;
	elseif (ew <= -4e-193)
		tmp = t_3;
	elseif (ew <= 2.15e-178)
		tmp = abs(Float64(Float64(tanh(asinh(Float64(Float64(tan(t) / ew) * Float64(-eh)))) * t_1) * eh));
	elseif (ew <= 3.1e+37)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = (-N[Sin[t], $MachinePrecision])}, Block[{t$95$2 = N[Abs[N[(N[Cos[N[ArcTan[N[(N[(t$95$1 / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * (-eh)), $MachinePrecision] + N[(N[Cos[N[ArcTan[N[(N[(N[(N[(eh / ew), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] * (-t)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.8e+81], t$95$2, If[LessEqual[ew, -4e-193], t$95$3, If[LessEqual[ew, 2.15e-178], N[Abs[N[(N[(N[Tanh[N[ArcSinh[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 3.1e+37], t$95$3, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -2.79999999999999995e81 or 3.1000000000000002e37 < ew

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 90.5%

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

   if -2.79999999999999995e81 < ew < -4.0000000000000002e-193 or 2.15e-178 < ew < 3.1000000000000002e37

   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. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.3%

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

   7. Applied rewrites 95.5%

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

   9. Applied rewrites 95.5%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 83.4%

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

   if -4.0000000000000002e-193 < ew < 2.15e-178

   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. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

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

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

   5. Applied rewrites 87.1%

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

   7. Applied rewrites 87.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.8 \cdot 10^{+81}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \mathbf{elif}\;ew \leq -4 \cdot 10^{-193}:\\ \;\;\;\;\left|\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{\tan t \cdot eh}{-ew}\right), \frac{\sin t}{ew} \cdot \left(-eh\right), \cos \tan^{-1} \left(\mathsf{fma}\left(\frac{eh}{ew} \cdot 0.3333333333333333, t \cdot t, \frac{eh}{ew}\right) \cdot \left(-t\right)\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{elif}\;ew \leq 2.15 \cdot 10^{-178}:\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right) \cdot \left(-\sin t\right)\right) \cdot eh\right|\\ \mathbf{elif}\;ew \leq 3.1 \cdot 10^{+37}:\\ \;\;\;\;\left|\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{\tan t \cdot eh}{-ew}\right), \frac{\sin t}{ew} \cdot \left(-eh\right), \cos \tan^{-1} \left(\mathsf{fma}\left(\frac{eh}{ew} \cdot 0.3333333333333333, t \cdot t, \frac{eh}{ew}\right) \cdot \left(-t\right)\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin t\\ \mathbf{if}\;ew \leq -2.9 \cdot 10^{-93} \lor \neg \left(ew \leq 2.7 \cdot 10^{+16}\right):\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{t\_1}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right) \cdot t\_1\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (- (sin t))))
   (if (or (<= ew -2.9e-93) (not (<= ew 2.7e+16)))
     (fabs (* (cos (atan (* (/ t_1 ew) (/ eh (cos t))))) (* (cos t) ew)))
     (fabs (* (* (tanh (asinh (* (/ (tan t) ew) (- eh)))) t_1) eh)))))

C

double code(double eh, double ew, double t) {
	double t_1 = -sin(t);
	double tmp;
	if ((ew <= -2.9e-93) || !(ew <= 2.7e+16)) {
		tmp = fabs((cos(atan(((t_1 / ew) * (eh / cos(t))))) * (cos(t) * ew)));
	} else {
		tmp = fabs(((tanh(asinh(((tan(t) / ew) * -eh))) * t_1) * eh));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = -math.sin(t)
	tmp = 0
	if (ew <= -2.9e-93) or not (ew <= 2.7e+16):
		tmp = math.fabs((math.cos(math.atan(((t_1 / ew) * (eh / math.cos(t))))) * (math.cos(t) * ew)))
	else:
		tmp = math.fabs(((math.tanh(math.asinh(((math.tan(t) / ew) * -eh))) * t_1) * eh))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(-sin(t))
	tmp = 0.0
	if ((ew <= -2.9e-93) || !(ew <= 2.7e+16))
		tmp = abs(Float64(cos(atan(Float64(Float64(t_1 / ew) * Float64(eh / cos(t))))) * Float64(cos(t) * ew)));
	else
		tmp = abs(Float64(Float64(tanh(asinh(Float64(Float64(tan(t) / ew) * Float64(-eh)))) * t_1) * eh));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = -sin(t);
	tmp = 0.0;
	if ((ew <= -2.9e-93) || ~((ew <= 2.7e+16)))
		tmp = abs((cos(atan(((t_1 / ew) * (eh / cos(t))))) * (cos(t) * ew)));
	else
		tmp = abs(((tanh(asinh(((tan(t) / ew) * -eh))) * t_1) * eh));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = (-N[Sin[t], $MachinePrecision])}, If[Or[LessEqual[ew, -2.9e-93], N[Not[LessEqual[ew, 2.7e+16]], $MachinePrecision]], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(t$95$1 / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Tanh[N[ArcSinh[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.8999999999999998e-93 or 2.7e16 < ew

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 83.0%

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

   if -2.8999999999999998e-93 < ew < 2.7e16

   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. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

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

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 75.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.9 \cdot 10^{-93} \lor \neg \left(ew \leq 2.7 \cdot 10^{+16}\right):\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right) \cdot \left(-\sin t\right)\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -2.9 \cdot 10^{-18}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{-eh}{\mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.041666666666666664 - 0.5, t \cdot t, 1\right)}\right) \cdot ew\right|\\ \mathbf{elif}\;ew \leq 2.75 \cdot 10^{+16}:\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right) \cdot \left(-\sin t\right)\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos t \cdot ew + \left(\sin t \cdot eh\right) \cdot \left(\frac{eh}{ew} \cdot \tan t\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -2.9e-18)
   (fabs
    (*
     (cos
      (atan
       (*
        (/ (sin t) ew)
        (/
         (- eh)
         (fma (- (* (* t t) 0.041666666666666664) 0.5) (* t t) 1.0)))))
     ew))
   (if (<= ew 2.75e+16)
     (fabs (* (* (tanh (asinh (* (/ (tan t) ew) (- eh)))) (- (sin t))) eh))
     (/ (+ (* (cos t) ew) (* (* (sin t) eh) (* (/ eh ew) (tan t)))) 1.0))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -2.9e-18) {
		tmp = fabs((cos(atan(((sin(t) / ew) * (-eh / fma((((t * t) * 0.041666666666666664) - 0.5), (t * t), 1.0))))) * ew));
	} else if (ew <= 2.75e+16) {
		tmp = fabs(((tanh(asinh(((tan(t) / ew) * -eh))) * -sin(t)) * eh));
	} else {
		tmp = ((cos(t) * ew) + ((sin(t) * eh) * ((eh / ew) * tan(t)))) / 1.0;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -2.9e-18)
		tmp = abs(Float64(cos(atan(Float64(Float64(sin(t) / ew) * Float64(Float64(-eh) / fma(Float64(Float64(Float64(t * t) * 0.041666666666666664) - 0.5), Float64(t * t), 1.0))))) * ew));
	elseif (ew <= 2.75e+16)
		tmp = abs(Float64(Float64(tanh(asinh(Float64(Float64(tan(t) / ew) * Float64(-eh)))) * Float64(-sin(t))) * eh));
	else
		tmp = Float64(Float64(Float64(cos(t) * ew) + Float64(Float64(sin(t) * eh) * Float64(Float64(eh / ew) * tan(t)))) / 1.0);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -2.9e-18], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * N[((-eh) / N[(N[(N[(N[(t * t), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 2.75e+16], N[Abs[N[(N[(N[Tanh[N[ArcSinh[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -2.9e-18

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 52.6%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{1 + {t}^{2} \cdot \left(\frac{1}{24} \cdot {t}^{2} - \frac{1}{2}\right)}\right) \cdot ew\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 52.6%

      \[\leadsto \left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.041666666666666664 - 0.5, t \cdot t, 1\right)}\right) \cdot ew\right| \]

   if -2.9e-18 < ew < 2.75e16

   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. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

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

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

   5. Applied rewrites 71.9%

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

   7. Applied rewrites 71.8%

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

   if 2.75e16 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 63.1%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 60.8%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.9 \cdot 10^{-18}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{-eh}{\mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.041666666666666664 - 0.5, t \cdot t, 1\right)}\right) \cdot ew\right|\\ \mathbf{elif}\;ew \leq 2.75 \cdot 10^{+16}:\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right) \cdot \left(-\sin t\right)\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos t \cdot ew + \left(\sin t \cdot eh\right) \cdot \left(\frac{eh}{ew} \cdot \tan t\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -2.9 \cdot 10^{-18}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{-eh}{\mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.041666666666666664 - 0.5, t \cdot t, 1\right)}\right) \cdot ew\right|\\ \mathbf{elif}\;ew \leq 2.75 \cdot 10^{+16}:\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(0.3333333333333333 \cdot \frac{eh}{ew}, t \cdot t, \frac{eh}{ew}\right) \cdot \left(-t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos t \cdot ew + \left(\sin t \cdot eh\right) \cdot \left(\frac{eh}{ew} \cdot \tan t\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -2.9e-18)
   (fabs
    (*
     (cos
      (atan
       (*
        (/ (sin t) ew)
        (/
         (- eh)
         (fma (- (* (* t t) 0.041666666666666664) 0.5) (* t t) 1.0)))))
     ew))
   (if (<= ew 2.75e+16)
     (fabs
      (*
       (* (- (sin t)) eh)
       (sin
        (atan
         (* (fma (* 0.3333333333333333 (/ eh ew)) (* t t) (/ eh ew)) (- t))))))
     (/ (+ (* (cos t) ew) (* (* (sin t) eh) (* (/ eh ew) (tan t)))) 1.0))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -2.9e-18) {
		tmp = fabs((cos(atan(((sin(t) / ew) * (-eh / fma((((t * t) * 0.041666666666666664) - 0.5), (t * t), 1.0))))) * ew));
	} else if (ew <= 2.75e+16) {
		tmp = fabs(((-sin(t) * eh) * sin(atan((fma((0.3333333333333333 * (eh / ew)), (t * t), (eh / ew)) * -t)))));
	} else {
		tmp = ((cos(t) * ew) + ((sin(t) * eh) * ((eh / ew) * tan(t)))) / 1.0;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -2.9e-18)
		tmp = abs(Float64(cos(atan(Float64(Float64(sin(t) / ew) * Float64(Float64(-eh) / fma(Float64(Float64(Float64(t * t) * 0.041666666666666664) - 0.5), Float64(t * t), 1.0))))) * ew));
	elseif (ew <= 2.75e+16)
		tmp = abs(Float64(Float64(Float64(-sin(t)) * eh) * sin(atan(Float64(fma(Float64(0.3333333333333333 * Float64(eh / ew)), Float64(t * t), Float64(eh / ew)) * Float64(-t))))));
	else
		tmp = Float64(Float64(Float64(cos(t) * ew) + Float64(Float64(sin(t) * eh) * Float64(Float64(eh / ew) * tan(t)))) / 1.0);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -2.9e-18], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * N[((-eh) / N[(N[(N[(N[(t * t), $MachinePrecision] * 0.041666666666666664), $MachinePrecision] - 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 2.75e+16], N[Abs[N[(N[((-N[Sin[t], $MachinePrecision]) * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[(0.3333333333333333 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] * (-t)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -2.9e-18

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 52.6%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{1 + {t}^{2} \cdot \left(\frac{1}{24} \cdot {t}^{2} - \frac{1}{2}\right)}\right) \cdot ew\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 52.6%

      \[\leadsto \left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.041666666666666664 - 0.5, t \cdot t, 1\right)}\right) \cdot ew\right| \]

   if -2.9e-18 < ew < 2.75e16

   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. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

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

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 68.2%

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

   if 2.75e16 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 63.1%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 60.8%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.9 \cdot 10^{-18}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{-eh}{\mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.041666666666666664 - 0.5, t \cdot t, 1\right)}\right) \cdot ew\right|\\ \mathbf{elif}\;ew \leq 2.75 \cdot 10^{+16}:\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(0.3333333333333333 \cdot \frac{eh}{ew}, t \cdot t, \frac{eh}{ew}\right) \cdot \left(-t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos t \cdot ew + \left(\sin t \cdot eh\right) \cdot \left(\frac{eh}{ew} \cdot \tan t\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.7% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \cos t \cdot ew \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[(N[Cos[t], $MachinePrecision] * ew), $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 40.6%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. distribute-rgt-out N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 14.4

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

7. Applied rewrites 14.4%

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

8. Taylor expanded in eh around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 31.1

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

10. Applied rewrites 31.1%

    \[\leadsto \color{blue}{\cos t \cdot ew} \]
11. Add Preprocessing

Alternative 14: 19.5% accurate, 50.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[(N[(N[(-0.5 * ew), $MachinePrecision] * t), $MachinePrecision] * t + ew), $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 40.6%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. distribute-rgt-out N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 14.4

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

7. Applied rewrites 14.4%

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

8. Taylor expanded in eh around 0

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

10. Applied rewrites 16.2%

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

12. Applied rewrites 16.2%

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

Alternative 15: 19.4% accurate, 50.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[(N[(-0.5 * ew), $MachinePrecision] * N[(t * t), $MachinePrecision] + ew), $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 40.6%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. distribute-rgt-out N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 14.4

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

7. Applied rewrites 14.4%

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

8. Taylor expanded in eh around 0

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

10. Applied rewrites 16.2%

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

Reproduce

herbie shell --seed 2024332 
(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)))))))