Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 21.4s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in ew around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-atan.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-tan.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

5. Applied rewrites 99.8%

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

Alternative 2: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\\ t_2 := \frac{1}{\left|\frac{1}{\mathsf{fma}\left(eh, \cos t \cdot \sin t\_1, \frac{ew \cdot \sin t}{\sqrt{\mathsf{fma}\left({\tan t}^{-2}, \frac{eh}{ew} \cdot \frac{eh}{ew}, 1\right)}}\right)}\right|}\\ \mathbf{if}\;t \leq -0.0003:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 0.005:\\ \;\;\;\;\left|t \cdot \left(\cos t\_1 \cdot \mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot -0.16666666666666666, ew\right)\right) + \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(eh \cdot \left(t \cdot t\right), 0.041666666666666664, eh \cdot -0.5\right), eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ eh (* ew (tan t)))))
        (t_2
         (/
          1.0
          (fabs
           (/
            1.0
            (fma
             eh
             (* (cos t) (sin t_1))
             (/
              (* ew (sin t))
              (sqrt
               (fma (pow (tan t) -2.0) (* (/ eh ew) (/ eh ew)) 1.0)))))))))
   (if (<= t -0.0003)
     t_2
     (if (<= t 0.005)
       (fabs
        (+
         (* t (* (cos t_1) (fma ew (* (* t t) -0.16666666666666666) ew)))
         (*
          (fma
           (* t t)
           (fma (* eh (* t t)) 0.041666666666666664 (* eh -0.5))
           eh)
          (sin (atan (/ (/ eh ew) (tan t)))))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((eh / (ew * tan(t))));
	double t_2 = 1.0 / fabs((1.0 / fma(eh, (cos(t) * sin(t_1)), ((ew * sin(t)) / sqrt(fma(pow(tan(t), -2.0), ((eh / ew) * (eh / ew)), 1.0))))));
	double tmp;
	if (t <= -0.0003) {
		tmp = t_2;
	} else if (t <= 0.005) {
		tmp = fabs(((t * (cos(t_1) * fma(ew, ((t * t) * -0.16666666666666666), ew))) + (fma((t * t), fma((eh * (t * t)), 0.041666666666666664, (eh * -0.5)), eh) * sin(atan(((eh / ew) / tan(t)))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(eh / Float64(ew * tan(t))))
	t_2 = Float64(1.0 / abs(Float64(1.0 / fma(eh, Float64(cos(t) * sin(t_1)), Float64(Float64(ew * sin(t)) / sqrt(fma((tan(t) ^ -2.0), Float64(Float64(eh / ew) * Float64(eh / ew)), 1.0)))))))
	tmp = 0.0
	if (t <= -0.0003)
		tmp = t_2;
	elseif (t <= 0.005)
		tmp = abs(Float64(Float64(t * Float64(cos(t_1) * fma(ew, Float64(Float64(t * t) * -0.16666666666666666), ew))) + Float64(fma(Float64(t * t), fma(Float64(eh * Float64(t * t)), 0.041666666666666664, Float64(eh * -0.5)), eh) * sin(atan(Float64(Float64(eh / ew) / tan(t)))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Abs[N[(1.0 / N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[Power[N[Tan[t], $MachinePrecision], -2.0], $MachinePrecision] * N[(N[(eh / ew), $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -0.0003], t$95$2, If[LessEqual[t, 0.005], N[Abs[N[(N[(t * N[(N[Cos[t$95$1], $MachinePrecision] * N[(ew * N[(N[(t * t), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t * t), $MachinePrecision] * N[(N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664 + N[(eh * -0.5), $MachinePrecision]), $MachinePrecision] + eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.99999999999999974e-4 or 0.0050000000000000001 < t

   1. Initial program 99.6%

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

   4. Applied rewrites 87.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-tan.f64 N/A

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

   6. Applied rewrites 99.5%

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

   if -2.99999999999999974e-4 < t < 0.0050000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \left|\color{blue}{t \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot -0.16666666666666666, ew\right)\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   6. Taylor expanded in t around 0

      \[\leadsto \left|t \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot \frac{-1}{6}, ew\right)\right) + \color{blue}{\left(eh + {t}^{2} \cdot \left(\frac{-1}{2} \cdot eh + \frac{1}{24} \cdot \left(eh \cdot {t}^{2}\right)\right)\right)} \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|t \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot \frac{-1}{6}, ew\right)\right) + \color{blue}{\left({t}^{2} \cdot \left(\frac{-1}{2} \cdot eh + \frac{1}{24} \cdot \left(eh \cdot {t}^{2}\right)\right) + eh\right)} \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      2. lower-fma.f64 N/A

         \[\leadsto \left|t \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot \frac{-1}{6}, ew\right)\right) + \color{blue}{\mathsf{fma}\left({t}^{2}, \frac{-1}{2} \cdot eh + \frac{1}{24} \cdot \left(eh \cdot {t}^{2}\right), eh\right)} \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      3. unpow2 N/A

         \[\leadsto \left|t \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot \frac{-1}{6}, ew\right)\right) + \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{-1}{2} \cdot eh + \frac{1}{24} \cdot \left(eh \cdot {t}^{2}\right), eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|t \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot \frac{-1}{6}, ew\right)\right) + \mathsf{fma}\left(\color{blue}{t \cdot t}, \frac{-1}{2} \cdot eh + \frac{1}{24} \cdot \left(eh \cdot {t}^{2}\right), eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      5. +-commutative N/A

         \[\leadsto \left|t \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot \frac{-1}{6}, ew\right)\right) + \mathsf{fma}\left(t \cdot t, \color{blue}{\frac{1}{24} \cdot \left(eh \cdot {t}^{2}\right) + \frac{-1}{2} \cdot eh}, eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      6. *-commutative N/A

         \[\leadsto \left|t \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot \frac{-1}{6}, ew\right)\right) + \mathsf{fma}\left(t \cdot t, \color{blue}{\left(eh \cdot {t}^{2}\right) \cdot \frac{1}{24}} + \frac{-1}{2} \cdot eh, eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      7. lower-fma.f64 N/A

         \[\leadsto \left|t \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot \frac{-1}{6}, ew\right)\right) + \mathsf{fma}\left(t \cdot t, \color{blue}{\mathsf{fma}\left(eh \cdot {t}^{2}, \frac{1}{24}, \frac{-1}{2} \cdot eh\right)}, eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|t \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot \frac{-1}{6}, ew\right)\right) + \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(\color{blue}{eh \cdot {t}^{2}}, \frac{1}{24}, \frac{-1}{2} \cdot eh\right), eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      9. unpow2 N/A

         \[\leadsto \left|t \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot \frac{-1}{6}, ew\right)\right) + \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(eh \cdot \color{blue}{\left(t \cdot t\right)}, \frac{1}{24}, \frac{-1}{2} \cdot eh\right), eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|t \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot \frac{-1}{6}, ew\right)\right) + \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(eh \cdot \color{blue}{\left(t \cdot t\right)}, \frac{1}{24}, \frac{-1}{2} \cdot eh\right), eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      11. *-commutative N/A

          \[\leadsto \left|t \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot \frac{-1}{6}, ew\right)\right) + \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(eh \cdot \left(t \cdot t\right), \frac{1}{24}, \color{blue}{eh \cdot \frac{-1}{2}}\right), eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

      12. lower-*.f64 100.0

          \[\leadsto \left|t \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot -0.16666666666666666, ew\right)\right) + \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(eh \cdot \left(t \cdot t\right), 0.041666666666666664, \color{blue}{eh \cdot -0.5}\right), eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]

   8. Applied rewrites 100.0%

      \[\leadsto \left|t \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \mathsf{fma}\left(ew, \left(t \cdot t\right) \cdot -0.16666666666666666, ew\right)\right) + \color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(eh \cdot \left(t \cdot t\right), 0.041666666666666664, eh \cdot -0.5\right), eh\right)} \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\\ t_2 := \left|eh \cdot \left(ew \cdot \mathsf{fma}\left(\cos t, \frac{t\_1}{ew}, \cos \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \frac{\sin t}{eh}\right)\right)\right|\\ \mathbf{if}\;eh \leq -1.52 \cdot 10^{-12}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 1.85 \cdot 10^{-91}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew \cdot \mathsf{fma}\left(eh, \frac{\cos t \cdot t\_1}{ew}, \sin t\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sin (atan (/ eh (* ew (tan t))))))
        (t_2
         (fabs
          (*
           eh
           (*
            ew
            (fma
             (cos t)
             (/ t_1 ew)
             (* (cos (atan (/ eh (* ew t)))) (/ (sin t) eh))))))))
   (if (<= eh -1.52e-12)
     t_2
     (if (<= eh 1.85e-91)
       (/ 1.0 (fabs (/ 1.0 (* ew (fma eh (/ (* (cos t) t_1) ew) (sin t))))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(atan((eh / (ew * tan(t)))));
	double t_2 = fabs((eh * (ew * fma(cos(t), (t_1 / ew), (cos(atan((eh / (ew * t)))) * (sin(t) / eh))))));
	double tmp;
	if (eh <= -1.52e-12) {
		tmp = t_2;
	} else if (eh <= 1.85e-91) {
		tmp = 1.0 / fabs((1.0 / (ew * fma(eh, ((cos(t) * t_1) / ew), sin(t)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = sin(atan(Float64(eh / Float64(ew * tan(t)))))
	t_2 = abs(Float64(eh * Float64(ew * fma(cos(t), Float64(t_1 / ew), Float64(cos(atan(Float64(eh / Float64(ew * t)))) * Float64(sin(t) / eh))))))
	tmp = 0.0
	if (eh <= -1.52e-12)
		tmp = t_2;
	elseif (eh <= 1.85e-91)
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(ew * fma(eh, Float64(Float64(cos(t) * t_1) / ew), sin(t))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(eh * N[(ew * N[(N[Cos[t], $MachinePrecision] * N[(t$95$1 / ew), $MachinePrecision] + N[(N[Cos[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] / eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.52e-12], t$95$2, If[LessEqual[eh, 1.85e-91], N[(1.0 / N[Abs[N[(1.0 / N[(ew * N[(eh * N[(N[(N[Cos[t], $MachinePrecision] * t$95$1), $MachinePrecision] / ew), $MachinePrecision] + N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.52e-12 or 1.8500000000000001e-91 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-tan.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in ew around inf

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

   11. Applied rewrites 99.6%

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

   12. Taylor expanded in t around 0

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

   14. Applied rewrites 99.6%

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

   if -1.52e-12 < eh < 1.8500000000000001e-91

   1. Initial program 99.8%

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

   4. Applied rewrites 79.2%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 98.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.52 \cdot 10^{-12}:\\ \;\;\;\;\left|eh \cdot \left(ew \cdot \mathsf{fma}\left(\cos t, \frac{\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}{ew}, \cos \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \frac{\sin t}{eh}\right)\right)\right|\\ \mathbf{elif}\;eh \leq 1.85 \cdot 10^{-91}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew \cdot \mathsf{fma}\left(eh, \frac{\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}{ew}, \sin t\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(ew \cdot \mathsf{fma}\left(\cos t, \frac{\sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}{ew}, \cos \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \frac{\sin t}{eh}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\left|\frac{1}{ew \cdot \mathsf{fma}\left(eh, \frac{\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)}{ew}, \sin t\right)}\right|}\\ \mathbf{if}\;ew \leq -4.2 \cdot 10^{-45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 6.6 \cdot 10^{-110}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (/
          1.0
          (fabs
           (/
            1.0
            (*
             ew
             (fma
              eh
              (/ (* (cos t) (sin (atan (/ eh (* ew (tan t)))))) ew)
              (sin t))))))))
   (if (<= ew -4.2e-45) t_1 (if (<= ew 6.6e-110) (fabs (* eh (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = 1.0 / fabs((1.0 / (ew * fma(eh, ((cos(t) * sin(atan((eh / (ew * tan(t)))))) / ew), sin(t)))));
	double tmp;
	if (ew <= -4.2e-45) {
		tmp = t_1;
	} else if (ew <= 6.6e-110) {
		tmp = fabs((eh * cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(1.0 / abs(Float64(1.0 / Float64(ew * fma(eh, Float64(Float64(cos(t) * sin(atan(Float64(eh / Float64(ew * tan(t)))))) / ew), sin(t))))))
	tmp = 0.0
	if (ew <= -4.2e-45)
		tmp = t_1;
	elseif (ew <= 6.6e-110)
		tmp = abs(Float64(eh * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(1.0 / N[Abs[N[(1.0 / N[(ew * N[(eh * N[(N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -4.2e-45], t$95$1, If[LessEqual[ew, 6.6e-110], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -4.1999999999999999e-45 or 6.5999999999999998e-110 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

   7. Applied rewrites 96.2%

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

   if -4.1999999999999999e-45 < ew < 6.5999999999999998e-110

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   4. Applied rewrites 6.8%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 92.2

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

   7. Applied rewrites 92.2%

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

Alternative 5: 84.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ t_2 := eh \cdot \cos t\\ t_3 := \left|t\_2\right|\\ t_4 := \left|\mathsf{fma}\left(t\_2 \cdot \frac{eh}{ew}, \frac{1}{\frac{eh}{ew}}, \frac{t\_1}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}\right)\right|\\ \mathbf{if}\;eh \leq -9 \cdot 10^{+64}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eh \leq -2.55 \cdot 10^{-160}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eh \leq 2.3 \cdot 10^{-136}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{elif}\;eh \leq 1.75 \cdot 10^{+34}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t)))
        (t_2 (* eh (cos t)))
        (t_3 (fabs t_2))
        (t_4
         (fabs
          (fma
           (* t_2 (/ eh ew))
           (/ 1.0 (/ eh ew))
           (/ t_1 (sqrt (fma (* eh eh) (pow (* ew (tan t)) -2.0) 1.0)))))))
   (if (<= eh -9e+64)
     t_3
     (if (<= eh -2.55e-160)
       t_4
       (if (<= eh 2.3e-136) (fabs t_1) (if (<= eh 1.75e+34) t_4 t_3))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double t_2 = eh * cos(t);
	double t_3 = fabs(t_2);
	double t_4 = fabs(fma((t_2 * (eh / ew)), (1.0 / (eh / ew)), (t_1 / sqrt(fma((eh * eh), pow((ew * tan(t)), -2.0), 1.0)))));
	double tmp;
	if (eh <= -9e+64) {
		tmp = t_3;
	} else if (eh <= -2.55e-160) {
		tmp = t_4;
	} else if (eh <= 2.3e-136) {
		tmp = fabs(t_1);
	} else if (eh <= 1.75e+34) {
		tmp = t_4;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	t_2 = Float64(eh * cos(t))
	t_3 = abs(t_2)
	t_4 = abs(fma(Float64(t_2 * Float64(eh / ew)), Float64(1.0 / Float64(eh / ew)), Float64(t_1 / sqrt(fma(Float64(eh * eh), (Float64(ew * tan(t)) ^ -2.0), 1.0)))))
	tmp = 0.0
	if (eh <= -9e+64)
		tmp = t_3;
	elseif (eh <= -2.55e-160)
		tmp = t_4;
	elseif (eh <= 2.3e-136)
		tmp = abs(t_1);
	elseif (eh <= 1.75e+34)
		tmp = t_4;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Abs[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[Abs[N[(N[(t$95$2 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(eh / ew), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 / N[Sqrt[N[(N[(eh * eh), $MachinePrecision] * N[Power[N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -9e+64], t$95$3, If[LessEqual[eh, -2.55e-160], t$95$4, If[LessEqual[eh, 2.3e-136], N[Abs[t$95$1], $MachinePrecision], If[LessEqual[eh, 1.75e+34], t$95$4, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -8.99999999999999946e64 or 1.74999999999999999e34 < eh

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   4. Applied rewrites 17.8%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 86.6

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

   7. Applied rewrites 86.6%

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

   if -8.99999999999999946e64 < eh < -2.55e-160 or 2.29999999999999998e-136 < eh < 1.74999999999999999e34

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   4. Applied rewrites 54.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 89.2

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

   7. Applied rewrites 89.2%

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

   if -2.55e-160 < eh < 2.29999999999999998e-136

   1. Initial program 99.8%

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

   4. Applied rewrites 19.5%

      \[\leadsto \left|\color{blue}{\frac{{\left(\frac{ew \cdot \sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}\right)}^{3} + {\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)}^{3}}{\mathsf{fma}\left(\left(eh \cdot eh\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right), \frac{ew \cdot \sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}} \cdot \frac{ew \cdot \sin t - \cos t \cdot \frac{eh \cdot eh}{ew \cdot \tan t}}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}\right)}}\right| \]

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 74.6

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

   7. Applied rewrites 74.6%

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

Alternative 6: 74.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;ew \leq -6.8 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= ew -6.8e+25) t_1 (if (<= ew 3.4e+39) (fabs (* eh (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (ew <= -6.8e+25) {
		tmp = t_1;
	} else if (ew <= 3.4e+39) {
		tmp = fabs((eh * cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((ew * sin(t)))
    if (ew <= (-6.8d+25)) then
        tmp = t_1
    else if (ew <= 3.4d+39) then
        tmp = abs((eh * cos(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * Math.sin(t)));
	double tmp;
	if (ew <= -6.8e+25) {
		tmp = t_1;
	} else if (ew <= 3.4e+39) {
		tmp = Math.abs((eh * Math.cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.sin(t)))
	tmp = 0
	if ew <= -6.8e+25:
		tmp = t_1
	elif ew <= 3.4e+39:
		tmp = math.fabs((eh * math.cos(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (ew <= -6.8e+25)
		tmp = t_1;
	elseif (ew <= 3.4e+39)
		tmp = abs(Float64(eh * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * sin(t)));
	tmp = 0.0;
	if (ew <= -6.8e+25)
		tmp = t_1;
	elseif (ew <= 3.4e+39)
		tmp = abs((eh * cos(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -6.8e+25], t$95$1, If[LessEqual[ew, 3.4e+39], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -6.79999999999999967e25 or 3.3999999999999999e39 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 33.3%

      \[\leadsto \left|\color{blue}{\frac{{\left(\frac{ew \cdot \sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}\right)}^{3} + {\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)}^{3}}{\mathsf{fma}\left(\left(eh \cdot eh\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right), \frac{ew \cdot \sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}} \cdot \frac{ew \cdot \sin t - \cos t \cdot \frac{eh \cdot eh}{ew \cdot \tan t}}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}\right)}}\right| \]

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 69.7

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

   7. Applied rewrites 69.7%

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

   if -6.79999999999999967e25 < ew < 3.3999999999999999e39

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   4. Applied rewrites 18.6%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 85.0

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

   7. Applied rewrites 85.0%

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

Alternative 7: 64.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \cos t\right|\\ \mathbf{if}\;eh \leq -2.05 \cdot 10^{-181}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 2.9 \cdot 10^{-147}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (cos t)))))
   (if (<= eh -2.05e-181) t_1 (if (<= eh 2.9e-147) (fabs (* ew t)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * cos(t)));
	double tmp;
	if (eh <= -2.05e-181) {
		tmp = t_1;
	} else if (eh <= 2.9e-147) {
		tmp = fabs((ew * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((eh * cos(t)))
    if (eh <= (-2.05d-181)) then
        tmp = t_1
    else if (eh <= 2.9d-147) then
        tmp = abs((ew * t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * Math.cos(t)));
	double tmp;
	if (eh <= -2.05e-181) {
		tmp = t_1;
	} else if (eh <= 2.9e-147) {
		tmp = Math.abs((ew * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.cos(t)))
	tmp = 0
	if eh <= -2.05e-181:
		tmp = t_1
	elif eh <= 2.9e-147:
		tmp = math.fabs((ew * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * cos(t)))
	tmp = 0.0
	if (eh <= -2.05e-181)
		tmp = t_1;
	elseif (eh <= 2.9e-147)
		tmp = abs(Float64(ew * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * cos(t)));
	tmp = 0.0;
	if (eh <= -2.05e-181)
		tmp = t_1;
	elseif (eh <= 2.9e-147)
		tmp = abs((ew * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.05e-181], t$95$1, If[LessEqual[eh, 2.9e-147], N[Abs[N[(ew * t), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.0500000000000001e-181 or 2.9000000000000001e-147 < eh

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   4. Applied rewrites 37.7%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 72.4

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

   7. Applied rewrites 72.4%

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

   if -2.0500000000000001e-181 < eh < 2.9000000000000001e-147

   1. Initial program 99.8%

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

   4. Applied rewrites 21.3%

      \[\leadsto \left|\color{blue}{\frac{{\left(\frac{ew \cdot \sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}\right)}^{3} + {\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)}^{3}}{\mathsf{fma}\left(\left(eh \cdot eh\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right), \frac{ew \cdot \sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}} \cdot \frac{ew \cdot \sin t - \cos t \cdot \frac{eh \cdot eh}{ew \cdot \tan t}}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}\right)}}\right| \]

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 78.8

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

   7. Applied rewrites 78.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 48.2%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.05 \cdot 10^{-181}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{elif}\;eh \leq 2.9 \cdot 10^{-147}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.8% accurate, 43.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot 1\right|\\ \mathbf{if}\;eh \leq -1.22 \cdot 10^{-170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 3.4 \cdot 10^{-147}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh 1.0))))
   (if (<= eh -1.22e-170) t_1 (if (<= eh 3.4e-147) (fabs (* ew t)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * 1.0));
	double tmp;
	if (eh <= -1.22e-170) {
		tmp = t_1;
	} else if (eh <= 3.4e-147) {
		tmp = fabs((ew * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((eh * 1.0d0))
    if (eh <= (-1.22d-170)) then
        tmp = t_1
    else if (eh <= 3.4d-147) then
        tmp = abs((ew * t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * 1.0));
	double tmp;
	if (eh <= -1.22e-170) {
		tmp = t_1;
	} else if (eh <= 3.4e-147) {
		tmp = Math.abs((ew * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * 1.0))
	tmp = 0
	if eh <= -1.22e-170:
		tmp = t_1
	elif eh <= 3.4e-147:
		tmp = math.fabs((ew * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * 1.0))
	tmp = 0.0
	if (eh <= -1.22e-170)
		tmp = t_1;
	elseif (eh <= 3.4e-147)
		tmp = abs(Float64(ew * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * 1.0));
	tmp = 0.0;
	if (eh <= -1.22e-170)
		tmp = t_1;
	elseif (eh <= 3.4e-147)
		tmp = abs((ew * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.22e-170], t$95$1, If[LessEqual[eh, 3.4e-147], N[Abs[N[(ew * t), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.22e-170 or 3.39999999999999996e-147 < eh

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

   4. Applied rewrites 37.4%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 73.3

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

   7. Applied rewrites 73.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 49.7%

       \[\leadsto \left|eh \cdot 1\right| \]

   if -1.22e-170 < eh < 3.39999999999999996e-147

   1. Initial program 99.8%

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

   4. Applied rewrites 21.5%

      \[\leadsto \left|\color{blue}{\frac{{\left(\frac{ew \cdot \sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}\right)}^{3} + {\left(eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right)}^{3}}{\mathsf{fma}\left(\left(eh \cdot eh\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right), \frac{ew \cdot \sin t}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}} \cdot \frac{ew \cdot \sin t - \cos t \cdot \frac{eh \cdot eh}{ew \cdot \tan t}}{\sqrt{\mathsf{fma}\left(eh \cdot eh, {\left(ew \cdot \tan t\right)}^{-2}, 1\right)}}\right)}}\right| \]

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 78.6

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

   7. Applied rewrites 78.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 46.9%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.22 \cdot 10^{-170}:\\ \;\;\;\;\left|eh \cdot 1\right|\\ \mathbf{elif}\;eh \leq 3.4 \cdot 10^{-147}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot 1\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.0% accurate, 108.8× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((eh * 1.0));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((eh * 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-atan.f64 N/A

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

   6. sin-atan N/A

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

   7. lift-/.f64 N/A

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

   8. div-inv N/A

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

   9. associate-/l* N/A

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

   10. associate-*r* N/A

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

4. Applied rewrites 42.5%

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

5. Taylor expanded in eh around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 62.8

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

7. Applied rewrites 62.8%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 43.2%

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

Reproduce

herbie shell --seed 2024234 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))