Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 15.2s

Alternatives: 13

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

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

FPCore

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

C

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

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(((sin(atan((eh / (tan(t) * ew)))) * (cos(t) * eh)) + ((sin(t) * ew) * cos(atan(((eh / ew) / tan(t)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 96.0% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (+
   (*
    (sin (atan (/ (* (fma -0.3333333333333333 (* t t) 1.0) (/ eh ew)) t)))
    (* (cos t) eh))
   (* (* (sin t) ew) (cos (atan (/ (/ eh ew) (tan t))))))))

C

double code(double eh, double ew, double t) {
	return fabs(((sin(atan(((fma(-0.3333333333333333, (t * t), 1.0) * (eh / ew)) / t))) * (cos(t) * eh)) + ((sin(t) * ew) * cos(atan(((eh / ew) / tan(t)))))));
}

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(sin(atan(Float64(Float64(fma(-0.3333333333333333, Float64(t * t), 1.0) * Float64(eh / ew)) / t))) * Float64(cos(t) * eh)) + Float64(Float64(sin(t) * ew) * cos(atan(Float64(Float64(eh / ew) / tan(t)))))))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[(N[(-0.3333333333333333 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $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 t around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

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

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 96.6%

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

6. Final simplification 96.6%

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

Alternative 3: 96.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left|\frac{\sin t \cdot ew}{\sqrt{{\left(\frac{ew}{\frac{eh}{\tan t}}\right)}^{-2} + 1}} + \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, t \cdot t, 1\right) \cdot \frac{eh}{ew}}{t}\right) \cdot \left(\cos t \cdot eh\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (+
   (/ (* (sin t) ew) (sqrt (+ (pow (/ ew (/ eh (tan t))) -2.0) 1.0)))
   (*
    (sin (atan (/ (* (fma -0.3333333333333333 (* t t) 1.0) (/ eh ew)) t)))
    (* (cos t) eh)))))

C

double code(double eh, double ew, double t) {
	return fabs((((sin(t) * ew) / sqrt((pow((ew / (eh / tan(t))), -2.0) + 1.0))) + (sin(atan(((fma(-0.3333333333333333, (t * t), 1.0) * (eh / ew)) / t))) * (cos(t) * eh))));
}

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(sin(t) * ew) / sqrt(Float64((Float64(ew / Float64(eh / tan(t))) ^ -2.0) + 1.0))) + Float64(sin(atan(Float64(Float64(fma(-0.3333333333333333, Float64(t * t), 1.0) * Float64(eh / ew)) / t))) * Float64(cos(t) * eh))))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[Sqrt[N[(N[Power[N[(ew / N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[N[ArcTan[N[(N[(N[(-0.3333333333333333 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $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 t around 0

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

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

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 96.6%

   \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(-0.3333333333333333, t \cdot t, 1\right) \cdot \frac{eh}{ew}}{t}\right)}\right| \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-atan.f64 N/A

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

   7. cos-atan N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

7. Applied rewrites 96.6%

   \[\leadsto \left|\color{blue}{\frac{\sin t \cdot ew}{\sqrt{{\left(\frac{ew}{\frac{eh}{\tan t}}\right)}^{-2} + 1}}} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, t \cdot t, 1\right) \cdot \frac{eh}{ew}}{t}\right)\right| \]

8. Final simplification 96.6%

   \[\leadsto \left|\frac{\sin t \cdot ew}{\sqrt{{\left(\frac{ew}{\frac{eh}{\tan t}}\right)}^{-2} + 1}} + \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, t \cdot t, 1\right) \cdot \frac{eh}{ew}}{t}\right) \cdot \left(\cos t \cdot eh\right)\right| \]
9. Add Preprocessing

Alternative 4: 75.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right)\\ t_2 := \cos t \cdot eh\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{+59}:\\ \;\;\;\;\left|\cos t\_1 \cdot \left(\sin t \cdot ew\right)\right|\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+38}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot t\_2 + \left(t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t\_1 \cdot t\_2\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (* (/ (/ eh (sin t)) ew) (cos t)))) (t_2 (* (cos t) eh)))
   (if (<= t -1.25e+59)
     (fabs (* (cos t_1) (* (sin t) ew)))
     (if (<= t 1.2e+38)
       (fabs
        (+
         (* (sin (atan (/ eh (* t ew)))) t_2)
         (* (* t ew) (cos (atan (/ (/ eh ew) (tan t)))))))
       (fabs (* (sin t_1) t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((((eh / sin(t)) / ew) * cos(t)));
	double t_2 = cos(t) * eh;
	double tmp;
	if (t <= -1.25e+59) {
		tmp = fabs((cos(t_1) * (sin(t) * ew)));
	} else if (t <= 1.2e+38) {
		tmp = fabs(((sin(atan((eh / (t * ew)))) * t_2) + ((t * ew) * cos(atan(((eh / ew) / tan(t)))))));
	} else {
		tmp = fabs((sin(t_1) * t_2));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = atan((((eh / sin(t)) / ew) * cos(t)))
    t_2 = cos(t) * eh
    if (t <= (-1.25d+59)) then
        tmp = abs((cos(t_1) * (sin(t) * ew)))
    else if (t <= 1.2d+38) then
        tmp = abs(((sin(atan((eh / (t * ew)))) * t_2) + ((t * ew) * cos(atan(((eh / ew) / tan(t)))))))
    else
        tmp = abs((sin(t_1) * t_2))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan((((eh / Math.sin(t)) / ew) * Math.cos(t)));
	double t_2 = Math.cos(t) * eh;
	double tmp;
	if (t <= -1.25e+59) {
		tmp = Math.abs((Math.cos(t_1) * (Math.sin(t) * ew)));
	} else if (t <= 1.2e+38) {
		tmp = Math.abs(((Math.sin(Math.atan((eh / (t * ew)))) * t_2) + ((t * ew) * Math.cos(Math.atan(((eh / ew) / Math.tan(t)))))));
	} else {
		tmp = Math.abs((Math.sin(t_1) * t_2));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((((eh / math.sin(t)) / ew) * math.cos(t)))
	t_2 = math.cos(t) * eh
	tmp = 0
	if t <= -1.25e+59:
		tmp = math.fabs((math.cos(t_1) * (math.sin(t) * ew)))
	elif t <= 1.2e+38:
		tmp = math.fabs(((math.sin(math.atan((eh / (t * ew)))) * t_2) + ((t * ew) * math.cos(math.atan(((eh / ew) / math.tan(t)))))))
	else:
		tmp = math.fabs((math.sin(t_1) * t_2))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(Float64(eh / sin(t)) / ew) * cos(t)))
	t_2 = Float64(cos(t) * eh)
	tmp = 0.0
	if (t <= -1.25e+59)
		tmp = abs(Float64(cos(t_1) * Float64(sin(t) * ew)));
	elseif (t <= 1.2e+38)
		tmp = abs(Float64(Float64(sin(atan(Float64(eh / Float64(t * ew)))) * t_2) + Float64(Float64(t * ew) * cos(atan(Float64(Float64(eh / ew) / tan(t)))))));
	else
		tmp = abs(Float64(sin(t_1) * t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((((eh / sin(t)) / ew) * cos(t)));
	t_2 = cos(t) * eh;
	tmp = 0.0;
	if (t <= -1.25e+59)
		tmp = abs((cos(t_1) * (sin(t) * ew)));
	elseif (t <= 1.2e+38)
		tmp = abs(((sin(atan((eh / (t * ew)))) * t_2) + ((t * ew) * cos(atan(((eh / ew) / tan(t)))))));
	else
		tmp = abs((sin(t_1) * t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[t, -1.25e+59], N[Abs[N[(N[Cos[t$95$1], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.2e+38], N[Abs[N[(N[(N[Sin[N[ArcTan[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[(t * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[t$95$1], $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.2499999999999999e59

   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

   3. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 60.1%

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

   if -1.2499999999999999e59 < t < 1.20000000000000009e38

   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. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 94.0

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

   8. Applied rewrites 94.0%

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

   if 1.20000000000000009e38 < 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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. *-commutative N/A

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

   5. Applied rewrites 57.4%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+59}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+38}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) \cdot \left(\cos t \cdot eh\right) + \left(t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right)\\ t_2 := \left|\sin t\_1 \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{if}\;eh \leq -1.42 \cdot 10^{-24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 6.7 \cdot 10^{-171}:\\ \;\;\;\;\left|\cos t\_1 \cdot \left(\sin t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (* (/ (/ eh (sin t)) ew) (cos t))))
        (t_2 (fabs (* (sin t_1) (* (cos t) eh)))))
   (if (<= eh -1.42e-24)
     t_2
     (if (<= eh 6.7e-171) (fabs (* (cos t_1) (* (sin t) ew))) t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((((eh / sin(t)) / ew) * cos(t)));
	double t_2 = fabs((sin(t_1) * (cos(t) * eh)));
	double tmp;
	if (eh <= -1.42e-24) {
		tmp = t_2;
	} else if (eh <= 6.7e-171) {
		tmp = fabs((cos(t_1) * (sin(t) * ew)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = atan((((eh / sin(t)) / ew) * cos(t)))
    t_2 = abs((sin(t_1) * (cos(t) * eh)))
    if (eh <= (-1.42d-24)) then
        tmp = t_2
    else if (eh <= 6.7d-171) then
        tmp = abs((cos(t_1) * (sin(t) * ew)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan((((eh / Math.sin(t)) / ew) * Math.cos(t)));
	double t_2 = Math.abs((Math.sin(t_1) * (Math.cos(t) * eh)));
	double tmp;
	if (eh <= -1.42e-24) {
		tmp = t_2;
	} else if (eh <= 6.7e-171) {
		tmp = Math.abs((Math.cos(t_1) * (Math.sin(t) * ew)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((((eh / math.sin(t)) / ew) * math.cos(t)))
	t_2 = math.fabs((math.sin(t_1) * (math.cos(t) * eh)))
	tmp = 0
	if eh <= -1.42e-24:
		tmp = t_2
	elif eh <= 6.7e-171:
		tmp = math.fabs((math.cos(t_1) * (math.sin(t) * ew)))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(Float64(eh / sin(t)) / ew) * cos(t)))
	t_2 = abs(Float64(sin(t_1) * Float64(cos(t) * eh)))
	tmp = 0.0
	if (eh <= -1.42e-24)
		tmp = t_2;
	elseif (eh <= 6.7e-171)
		tmp = abs(Float64(cos(t_1) * Float64(sin(t) * ew)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((((eh / sin(t)) / ew) * cos(t)));
	t_2 = abs((sin(t_1) * (cos(t) * eh)));
	tmp = 0.0;
	if (eh <= -1.42e-24)
		tmp = t_2;
	elseif (eh <= 6.7e-171)
		tmp = abs((cos(t_1) * (sin(t) * ew)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[Sin[t$95$1], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.42e-24], t$95$2, If[LessEqual[eh, 6.7e-171], N[Abs[N[(N[Cos[t$95$1], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.42e-24 or 6.69999999999999961e-171 < 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 eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. *-commutative N/A

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

   5. Applied rewrites 80.6%

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

   if -1.42e-24 < eh < 6.69999999999999961e-171

   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. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 74.8%

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

Alternative 6: 60.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(eh, 0.0011904761904761906, -0.0032407407407407406 \cdot eh\right) \cdot t, t, -0.019444444444444445 \cdot eh\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right|\\ \mathbf{if}\;eh \leq -2.8 \cdot 10^{-24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 4.4 \cdot 10^{-21}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           (sin
            (atan
             (*
              (/
               (/
                (fma
                 (fma
                  (-
                   (fma
                    (*
                     (fma
                      eh
                      0.0011904761904761906
                      (* -0.0032407407407407406 eh))
                     t)
                    t
                    (* -0.019444444444444445 eh)))
                  (* t t)
                  (* 0.16666666666666666 eh))
                 (* t t)
                 eh)
                t)
               ew)
              (cos t))))
           eh))))
   (if (<= eh -2.8e-24)
     t_1
     (if (<= eh 4.4e-21)
       (fabs (* (cos (atan (* (/ (/ eh (sin t)) ew) (cos t)))) (* (sin t) ew)))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(atan((((fma(fma(-fma((fma(eh, 0.0011904761904761906, (-0.0032407407407407406 * eh)) * t), t, (-0.019444444444444445 * eh)), (t * t), (0.16666666666666666 * eh)), (t * t), eh) / t) / ew) * cos(t)))) * eh));
	double tmp;
	if (eh <= -2.8e-24) {
		tmp = t_1;
	} else if (eh <= 4.4e-21) {
		tmp = fabs((cos(atan((((eh / sin(t)) / ew) * cos(t)))) * (sin(t) * ew)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(atan(Float64(Float64(Float64(fma(fma(Float64(-fma(Float64(fma(eh, 0.0011904761904761906, Float64(-0.0032407407407407406 * eh)) * t), t, Float64(-0.019444444444444445 * eh))), Float64(t * t), Float64(0.16666666666666666 * eh)), Float64(t * t), eh) / t) / ew) * cos(t)))) * eh))
	tmp = 0.0
	if (eh <= -2.8e-24)
		tmp = t_1;
	elseif (eh <= 4.4e-21)
		tmp = abs(Float64(cos(atan(Float64(Float64(Float64(eh / sin(t)) / ew) * cos(t)))) * Float64(sin(t) * ew)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[((-N[(N[(N[(eh * 0.0011904761904761906 + N[(-0.0032407407407407406 * eh), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * t + N[(-0.019444444444444445 * eh), $MachinePrecision]), $MachinePrecision]) * N[(t * t), $MachinePrecision] + N[(0.16666666666666666 * eh), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + eh), $MachinePrecision] / t), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.8e-24], t$95$1, If[LessEqual[eh, 4.4e-21], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.8000000000000002e-24 or 4.4000000000000001e-21 < 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 t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 59.7

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

   5. Applied rewrites 59.7%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{eh + {t}^{2} \cdot \left({t}^{2} \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(\frac{-1}{5040} \cdot eh + \left(\frac{1}{720} \cdot eh + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot eh + \frac{1}{120} \cdot eh\right)\right)\right)\right) - \left(\frac{-1}{36} \cdot eh + \frac{1}{120} \cdot eh\right)\right) - \frac{-1}{6} \cdot eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 59.9%

      \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(eh, 0.0011904761904761906, eh \cdot -0.0032407407407407406\right) \cdot t, t, eh \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]

   if -2.8000000000000002e-24 < eh < 4.4000000000000001e-21

   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 eh around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 69.5%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.8 \cdot 10^{-24}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(eh, 0.0011904761904761906, -0.0032407407407407406 \cdot eh\right) \cdot t, t, -0.019444444444444445 \cdot eh\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right|\\ \mathbf{elif}\;eh \leq 4.4 \cdot 10^{-21}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(eh, 0.0011904761904761906, -0.0032407407407407406 \cdot eh\right) \cdot t, t, -0.019444444444444445 \cdot eh\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(eh, 0.0011904761904761906, -0.0032407407407407406 \cdot eh\right) \cdot t, t, -0.019444444444444445 \cdot eh\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan
     (*
      (/
       (/
        (fma
         (fma
          (-
           (fma
            (* (fma eh 0.0011904761904761906 (* -0.0032407407407407406 eh)) t)
            t
            (* -0.019444444444444445 eh)))
          (* t t)
          (* 0.16666666666666666 eh))
         (* t t)
         eh)
        t)
       ew)
      (cos t))))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((((fma(fma(-fma((fma(eh, 0.0011904761904761906, (-0.0032407407407407406 * eh)) * t), t, (-0.019444444444444445 * eh)), (t * t), (0.16666666666666666 * eh)), (t * t), eh) / t) / ew) * cos(t)))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(Float64(fma(fma(Float64(-fma(Float64(fma(eh, 0.0011904761904761906, Float64(-0.0032407407407407406 * eh)) * t), t, Float64(-0.019444444444444445 * eh))), Float64(t * t), Float64(0.16666666666666666 * eh)), Float64(t * t), eh) / t) / ew) * cos(t)))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[((-N[(N[(N[(eh * 0.0011904761904761906 + N[(-0.0032407407407407406 * eh), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * t + N[(-0.019444444444444445 * eh), $MachinePrecision]), $MachinePrecision]) * N[(t * t), $MachinePrecision] + N[(0.16666666666666666 * eh), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + eh), $MachinePrecision] / t), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $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 t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 44.8

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

5. Applied rewrites 44.8%

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

6. Taylor expanded in t around 0

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{eh + {t}^{2} \cdot \left({t}^{2} \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(\frac{-1}{5040} \cdot eh + \left(\frac{1}{720} \cdot eh + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot eh + \frac{1}{120} \cdot eh\right)\right)\right)\right) - \left(\frac{-1}{36} \cdot eh + \frac{1}{120} \cdot eh\right)\right) - \frac{-1}{6} \cdot eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]
7. Step-by-step derivation

8. Applied rewrites 45.0%

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(eh, 0.0011904761904761906, eh \cdot -0.0032407407407407406\right) \cdot t, t, eh \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]

9. Final simplification 45.0%

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-\mathsf{fma}\left(\mathsf{fma}\left(eh, 0.0011904761904761906, -0.0032407407407407406 \cdot eh\right) \cdot t, t, -0.019444444444444445 \cdot eh\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]
10. Add Preprocessing

Alternative 8: 42.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445 \cdot eh, t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan
     (*
      (/
       (/
        (fma
         (fma (* 0.019444444444444445 eh) (* t t) (* 0.16666666666666666 eh))
         (* t t)
         eh)
        t)
       ew)
      (cos t))))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((((fma(fma((0.019444444444444445 * eh), (t * t), (0.16666666666666666 * eh)), (t * t), eh) / t) / ew) * cos(t)))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(Float64(fma(fma(Float64(0.019444444444444445 * eh), Float64(t * t), Float64(0.16666666666666666 * eh)), Float64(t * t), eh) / t) / ew) * cos(t)))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(N[(0.019444444444444445 * eh), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(0.16666666666666666 * eh), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + eh), $MachinePrecision] / t), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $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 t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 44.8

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

5. Applied rewrites 44.8%

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

6. Taylor expanded in t around 0

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{eh + {t}^{2} \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(\frac{-1}{36} \cdot eh + \frac{1}{120} \cdot eh\right)\right) - \frac{-1}{6} \cdot eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]
7. Step-by-step derivation

8. Applied rewrites 45.0%

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(eh \cdot 0.019444444444444445, t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]

9. Final simplification 45.0%

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.019444444444444445 \cdot eh, t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]
10. Add Preprocessing

Alternative 9: 42.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(0.16666666666666666, \left(t \cdot t\right) \cdot eh, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan
     (* (/ (/ (fma 0.16666666666666666 (* (* t t) eh) eh) t) ew) (cos t))))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((((fma(0.16666666666666666, ((t * t) * eh), eh) / t) / ew) * cos(t)))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(Float64(fma(0.16666666666666666, Float64(Float64(t * t) * eh), eh) / t) / ew) * cos(t)))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(0.16666666666666666 * N[(N[(t * t), $MachinePrecision] * eh), $MachinePrecision] + eh), $MachinePrecision] / t), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $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 t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 44.8

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

5. Applied rewrites 44.8%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 45.0%

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(0.16666666666666666, \left(t \cdot t\right) \cdot eh, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]
9. Add Preprocessing

Alternative 10: 42.5% accurate, 3.4× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan (/ (* (fma (/ (* t t) ew) -0.3333333333333333 (/ 1.0 ew)) eh) t)))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan(((fma(((t * t) / ew), -0.3333333333333333, (1.0 / ew)) * eh) / t))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(fma(Float64(Float64(t * t) / ew), -0.3333333333333333, Float64(1.0 / ew)) * eh) / t))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision] * -0.3333333333333333 + N[(1.0 / ew), $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $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 t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 44.8

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

5. Applied rewrites 44.8%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 38.2%

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

9. Taylor expanded in eh around 0

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

11. Applied rewrites 45.0%

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

Alternative 11: 41.8% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin (atan (* (/ eh t) (/ (fma (* t t) -0.3333333333333333 1.0) ew))))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan(((eh / t) * (fma((t * t), -0.3333333333333333, 1.0) / ew)))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(eh / t) * Float64(fma(Float64(t * t), -0.3333333333333333, 1.0) / ew)))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(eh / t), $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $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 t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 44.8

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

5. Applied rewrites 44.8%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 38.2%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 42.9%

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

12. Taylor expanded in ew around 0

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

14. Applied rewrites 44.1%

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

15. Final simplification 44.1%

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

Alternative 12: 40.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $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 t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 44.8

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

5. Applied rewrites 44.8%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 38.2%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 42.9%

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

Alternative 13: 14.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{ew}}{t}\\ \left|\frac{t\_1}{\sqrt{{t\_1}^{2} + 1}} \cdot eh\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) t)))
   (fabs (* (/ t_1 (sqrt (+ (pow t_1 2.0) 1.0))) eh))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / t;
	return fabs(((t_1 / sqrt((pow(t_1, 2.0) + 1.0))) * eh));
}

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 = (eh / ew) / t
    code = abs(((t_1 / sqrt(((t_1 ** 2.0d0) + 1.0d0))) * eh))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / t;
	return Math.abs(((t_1 / Math.sqrt((Math.pow(t_1, 2.0) + 1.0))) * eh));
}

Python

def code(eh, ew, t):
	t_1 = (eh / ew) / t
	return math.fabs(((t_1 / math.sqrt((math.pow(t_1, 2.0) + 1.0))) * eh))

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / t)
	return abs(Float64(Float64(t_1 / sqrt(Float64((t_1 ^ 2.0) + 1.0))) * eh))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = (eh / ew) / t;
	tmp = abs(((t_1 / sqrt(((t_1 ^ 2.0) + 1.0))) * eh));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]}, N[Abs[N[(N[(t$95$1 / N[Sqrt[N[(N[Power[t$95$1, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * eh), $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 t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 44.8

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

5. Applied rewrites 44.8%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 38.2%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 42.9%

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

13. Applied rewrites 15.6%

    \[\leadsto \left|\frac{\frac{\frac{eh}{ew}}{t}}{\sqrt{{\left(\frac{\frac{eh}{ew}}{t}\right)}^{2} + 1}} \cdot eh\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(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))))))))