Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 13.1s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in t around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 99.2

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

9. Applied rewrites 99.2%

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

10. Final simplification 99.2%

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

Alternative 3: 76.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(\left(0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right) \cdot t\right)\right|\\ t_2 := \frac{t}{ew} \cdot eh\\ \mathbf{if}\;eh \leq -4.2 \cdot 10^{+191}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 2.7 \cdot 10^{+160}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{t\_2 \cdot eh}{\sqrt{1 + {t\_2}^{2}}}, \sin t, \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot \left(\cos t \cdot ew\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           (* (sin t) eh)
           (sin
            (atan
             (*
              (fma (* (* 0.3333333333333333 (/ eh ew)) t) t (/ eh ew))
              t))))))
        (t_2 (* (/ t ew) eh)))
   (if (<= eh -4.2e+191)
     t_1
     (if (<= eh 2.7e+160)
       (fabs
        (fma
         (/ (* t_2 eh) (sqrt (+ 1.0 (pow t_2 2.0))))
         (sin t)
         (* (cos (atan (* (/ (tan t) ew) eh))) (* (cos t) ew))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((sin(t) * eh) * sin(atan((fma(((0.3333333333333333 * (eh / ew)) * t), t, (eh / ew)) * t)))));
	double t_2 = (t / ew) * eh;
	double tmp;
	if (eh <= -4.2e+191) {
		tmp = t_1;
	} else if (eh <= 2.7e+160) {
		tmp = fabs(fma(((t_2 * eh) / sqrt((1.0 + pow(t_2, 2.0)))), sin(t), (cos(atan(((tan(t) / ew) * eh))) * (cos(t) * ew))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(sin(t) * eh) * sin(atan(Float64(fma(Float64(Float64(0.3333333333333333 * Float64(eh / ew)) * t), t, Float64(eh / ew)) * t)))))
	t_2 = Float64(Float64(t / ew) * eh)
	tmp = 0.0
	if (eh <= -4.2e+191)
		tmp = t_1;
	elseif (eh <= 2.7e+160)
		tmp = abs(fma(Float64(Float64(t_2 * eh) / sqrt(Float64(1.0 + (t_2 ^ 2.0)))), sin(t), Float64(cos(atan(Float64(Float64(tan(t) / ew) * eh))) * Float64(cos(t) * ew))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[(N[(0.3333333333333333 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * t + N[(eh / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[eh, -4.2e+191], t$95$1, If[LessEqual[eh, 2.7e+160], N[Abs[N[(N[(N[(t$95$2 * eh), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[t], $MachinePrecision] + N[(N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.2000000000000001e191 or 2.7e160 < eh

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 88.1

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

   9. Applied rewrites 88.1%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 88.2%

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

   if -4.2000000000000001e191 < eh < 2.7e160

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in t around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 99.2

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

   9. Applied rewrites 99.2%

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

       1. lift-*.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. lift-atan.f64 N/A

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

       4. sin-atan N/A

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

       5. associate-*l/ N/A

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

       6. lower-/.f64 N/A

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

   11. Applied rewrites 83.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.2 \cdot 10^{+191}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(\left(0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right) \cdot t\right)\right|\\ \mathbf{elif}\;eh \leq 2.7 \cdot 10^{+160}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\left(\frac{t}{ew} \cdot eh\right) \cdot eh}{\sqrt{1 + {\left(\frac{t}{ew} \cdot eh\right)}^{2}}}, \sin t, \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot \left(\cos t \cdot ew\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(\left(0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right) \cdot t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \left|t\_1 \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(\left(0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right) \cdot t\right)\right|\\ \mathbf{if}\;eh \leq -1.02 \cdot 10^{+135}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 2.5 \cdot 10^{+30}:\\ \;\;\;\;\left|\frac{\left(\frac{eh}{ew} \cdot \tan t\right) \cdot t\_1 + \cos t \cdot ew}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh))
        (t_2
         (fabs
          (*
           t_1
           (sin
            (atan
             (*
              (fma (* (* 0.3333333333333333 (/ eh ew)) t) t (/ eh ew))
              t)))))))
   (if (<= eh -1.02e+135)
     t_2
     (if (<= eh 2.5e+30)
       (fabs
        (/
         (+ (* (* (/ eh ew) (tan t)) t_1) (* (cos t) ew))
         (/ 1.0 (cos (atan (* (/ (tan t) ew) eh))))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = fabs((t_1 * sin(atan((fma(((0.3333333333333333 * (eh / ew)) * t), t, (eh / ew)) * t)))));
	double tmp;
	if (eh <= -1.02e+135) {
		tmp = t_2;
	} else if (eh <= 2.5e+30) {
		tmp = fabs((((((eh / ew) * tan(t)) * t_1) + (cos(t) * ew)) / (1.0 / cos(atan(((tan(t) / ew) * eh))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = abs(Float64(t_1 * sin(atan(Float64(fma(Float64(Float64(0.3333333333333333 * Float64(eh / ew)) * t), t, Float64(eh / ew)) * t)))))
	tmp = 0.0
	if (eh <= -1.02e+135)
		tmp = t_2;
	elseif (eh <= 2.5e+30)
		tmp = abs(Float64(Float64(Float64(Float64(Float64(eh / ew) * tan(t)) * t_1) + Float64(cos(t) * ew)) / Float64(1.0 / cos(atan(Float64(Float64(tan(t) / ew) * eh))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(N[(N[(N[(0.3333333333333333 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * t + N[(eh / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.02e+135], t$95$2, If[LessEqual[eh, 2.5e+30], N[Abs[N[(N[(N[(N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.01999999999999993e135 or 2.4999999999999999e30 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 74.3

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

   9. Applied rewrites 74.3%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 74.4%

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

   if -1.01999999999999993e135 < eh < 2.4999999999999999e30

   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 87.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.7

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

   6. Applied rewrites 87.7%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.02 \cdot 10^{+135}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(\left(0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right) \cdot t\right)\right|\\ \mathbf{elif}\;eh \leq 2.5 \cdot 10^{+30}:\\ \;\;\;\;\left|\frac{\left(\frac{eh}{ew} \cdot \tan t\right) \cdot \left(\sin t \cdot eh\right) + \cos t \cdot ew}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(\left(0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right) \cdot t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew} \cdot eh\\ t_2 := \left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(\left(0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right) \cdot t\right)\right|\\ \mathbf{if}\;eh \leq -1.02 \cdot 10^{+135}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 2.5 \cdot 10^{+30}:\\ \;\;\;\;\left|\mathsf{fma}\left(-ew, \cos t, \left(-\sin t\right) \cdot \left(t\_1 \cdot eh\right)\right) \cdot \cos \tan^{-1} t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ (tan t) ew) eh))
        (t_2
         (fabs
          (*
           (* (sin t) eh)
           (sin
            (atan
             (*
              (fma (* (* 0.3333333333333333 (/ eh ew)) t) t (/ eh ew))
              t)))))))
   (if (<= eh -1.02e+135)
     t_2
     (if (<= eh 2.5e+30)
       (fabs
        (* (fma (- ew) (cos t) (* (- (sin t)) (* t_1 eh))) (cos (atan t_1))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = (tan(t) / ew) * eh;
	double t_2 = fabs(((sin(t) * eh) * sin(atan((fma(((0.3333333333333333 * (eh / ew)) * t), t, (eh / ew)) * t)))));
	double tmp;
	if (eh <= -1.02e+135) {
		tmp = t_2;
	} else if (eh <= 2.5e+30) {
		tmp = fabs((fma(-ew, cos(t), (-sin(t) * (t_1 * eh))) * cos(atan(t_1))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(tan(t) / ew) * eh)
	t_2 = abs(Float64(Float64(sin(t) * eh) * sin(atan(Float64(fma(Float64(Float64(0.3333333333333333 * Float64(eh / ew)) * t), t, Float64(eh / ew)) * t)))))
	tmp = 0.0
	if (eh <= -1.02e+135)
		tmp = t_2;
	elseif (eh <= 2.5e+30)
		tmp = abs(Float64(fma(Float64(-ew), cos(t), Float64(Float64(-sin(t)) * Float64(t_1 * eh))) * cos(atan(t_1))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[(N[(0.3333333333333333 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * t + N[(eh / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.02e+135], t$95$2, If[LessEqual[eh, 2.5e+30], N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision] + N[((-N[Sin[t], $MachinePrecision]) * N[(t$95$1 * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.01999999999999993e135 or 2.4999999999999999e30 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 74.3

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

   9. Applied rewrites 74.3%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 74.4%

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

   if -1.01999999999999993e135 < eh < 2.4999999999999999e30

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 87.7%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.02 \cdot 10^{+135}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(\left(0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right) \cdot t\right)\right|\\ \mathbf{elif}\;eh \leq 2.5 \cdot 10^{+30}:\\ \;\;\;\;\left|\mathsf{fma}\left(-ew, \cos t, \left(-\sin t\right) \cdot \left(\left(\frac{\tan t}{ew} \cdot eh\right) \cdot eh\right)\right) \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(\left(0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right) \cdot t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(\left(0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right) \cdot t\right)\right|\\ \mathbf{if}\;eh \leq -1.02 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 1.25 \cdot 10^{+25}:\\ \;\;\;\;\left|\left(-\cos t\right) \cdot ew\right| \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           (* (sin t) eh)
           (sin
            (atan
             (*
              (fma (* (* 0.3333333333333333 (/ eh ew)) t) t (/ eh ew))
              t)))))))
   (if (<= eh -1.02e+135)
     t_1
     (if (<= eh 1.25e+25)
       (* (fabs (* (- (cos t)) ew)) (cos (atan (* (/ (tan t) ew) eh))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((sin(t) * eh) * sin(atan((fma(((0.3333333333333333 * (eh / ew)) * t), t, (eh / ew)) * t)))));
	double tmp;
	if (eh <= -1.02e+135) {
		tmp = t_1;
	} else if (eh <= 1.25e+25) {
		tmp = fabs((-cos(t) * ew)) * cos(atan(((tan(t) / ew) * eh)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(sin(t) * eh) * sin(atan(Float64(fma(Float64(Float64(0.3333333333333333 * Float64(eh / ew)) * t), t, Float64(eh / ew)) * t)))))
	tmp = 0.0
	if (eh <= -1.02e+135)
		tmp = t_1;
	elseif (eh <= 1.25e+25)
		tmp = Float64(abs(Float64(Float64(-cos(t)) * ew)) * cos(atan(Float64(Float64(tan(t) / ew) * eh))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[(N[(0.3333333333333333 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * t + N[(eh / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.02e+135], t$95$1, If[LessEqual[eh, 1.25e+25], N[(N[Abs[N[((-N[Cos[t], $MachinePrecision]) * ew), $MachinePrecision]], $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.01999999999999993e135 or 1.25000000000000006e25 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 74.3

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

   9. Applied rewrites 74.3%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 74.4%

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

   if -1.01999999999999993e135 < eh < 1.25000000000000006e25

   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 87.7%

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

   5. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 86.7

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

   7. Applied rewrites 86.7%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-inv N/A

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

      4. fabs-mul N/A

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

      5. fabs-div N/A

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

   9. Applied rewrites 86.7%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.02 \cdot 10^{+135}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(\left(0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right) \cdot t\right)\right|\\ \mathbf{elif}\;eh \leq 1.25 \cdot 10^{+25}:\\ \;\;\;\;\left|\left(-\cos t\right) \cdot ew\right| \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(\left(0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right) \cdot t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(\left(0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right) \cdot t\right)\right|\\ \mathbf{if}\;eh \leq -1.02 \cdot 10^{+135}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 8.6 \cdot 10^{+21}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           (* (sin t) eh)
           (sin
            (atan
             (*
              (fma (* (* 0.3333333333333333 (/ eh ew)) t) t (/ eh ew))
              t)))))))
   (if (<= eh -1.02e+135) t_1 (if (<= eh 8.6e+21) (fabs (/ ew 1.0)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((sin(t) * eh) * sin(atan((fma(((0.3333333333333333 * (eh / ew)) * t), t, (eh / ew)) * t)))));
	double tmp;
	if (eh <= -1.02e+135) {
		tmp = t_1;
	} else if (eh <= 8.6e+21) {
		tmp = fabs((ew / 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(sin(t) * eh) * sin(atan(Float64(fma(Float64(Float64(0.3333333333333333 * Float64(eh / ew)) * t), t, Float64(eh / ew)) * t)))))
	tmp = 0.0
	if (eh <= -1.02e+135)
		tmp = t_1;
	elseif (eh <= 8.6e+21)
		tmp = abs(Float64(ew / 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[(N[(0.3333333333333333 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * t + N[(eh / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.02e+135], t$95$1, If[LessEqual[eh, 8.6e+21], N[Abs[N[(ew / 1.0), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.01999999999999993e135 or 8.6e21 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 74.3

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

   9. Applied rewrites 74.3%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 74.4%

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

   if -1.01999999999999993e135 < eh < 8.6e21

   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 60.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 59.3%

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

   10. Applied rewrites 58.7%

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

   11. Taylor expanded in eh around 0

       \[\leadsto \left|\frac{ew}{1}\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 60.7%

       \[\leadsto \left|\frac{ew}{1}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.02 \cdot 10^{+135}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(\left(0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right) \cdot t\right)\right|\\ \mathbf{elif}\;eh \leq 8.6 \cdot 10^{+21}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(\left(0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right) \cdot t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.1% accurate, 61.6× speedup

Math

\[\begin{array}{l} \\ \left|\frac{ew}{1}\right| \end{array} \]

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((ew / 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((ew / 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew / 1.0), $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. 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 47.7%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 46.6%

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

10. Applied rewrites 45.8%

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

11. Taylor expanded in eh around 0

    \[\leadsto \left|\frac{ew}{1}\right| \]
12. Step-by-step derivation

13. Applied rewrites 47.8%

    \[\leadsto \left|\frac{ew}{1}\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024288 
(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)))))))