Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 23.0s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 2: 30.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{if}\;\left(ew \cdot \sin t\right) \cdot \cos t\_1 + \left(eh \cdot \cos t\right) \cdot \sin t\_1 \leq -4 \cdot 10^{-215}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \mathbf{else}:\\ \;\;\;\;\sin t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (/ eh ew) (tan t)))))
   (if (<=
        (+ (* (* ew (sin t)) (cos t_1)) (* (* eh (cos t)) (sin t_1)))
        -4e-215)
     (fabs (* ew t))
     (* (sin t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	double tmp;
	if ((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))) <= -4e-215) {
		tmp = fabs((ew * t));
	} else {
		tmp = sin(t) * ew;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan(((eh / ew) / tan(t)))
    if ((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))) <= (-4d-215)) then
        tmp = abs((ew * t))
    else
        tmp = sin(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh / ew) / Math.tan(t)));
	double tmp;
	if ((((ew * Math.sin(t)) * Math.cos(t_1)) + ((eh * Math.cos(t)) * Math.sin(t_1))) <= -4e-215) {
		tmp = Math.abs((ew * t));
	} else {
		tmp = Math.sin(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh / ew) / math.tan(t)))
	tmp = 0
	if (((ew * math.sin(t)) * math.cos(t_1)) + ((eh * math.cos(t)) * math.sin(t_1))) <= -4e-215:
		tmp = math.fabs((ew * t))
	else:
		tmp = math.sin(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh / ew) / tan(t)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * sin(t)) * cos(t_1)) + Float64(Float64(eh * cos(t)) * sin(t_1))) <= -4e-215)
		tmp = abs(Float64(ew * t));
	else
		tmp = Float64(sin(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((eh / ew) / tan(t)));
	tmp = 0.0;
	if ((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))) <= -4e-215)
		tmp = abs((ew * t));
	else
		tmp = sin(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -4e-215], N[Abs[N[(ew * t), $MachinePrecision]], $MachinePrecision], N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t)))))) < -4.00000000000000017e-215

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 60.4%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 39.6

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

   7. Applied rewrites 39.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 15.2%

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

   if -4.00000000000000017e-215 < (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))))

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 67.6%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 46.8

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

   7. Applied rewrites 46.8%

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

      1. lift-fabs.f64 N/A

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

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

         \[\leadsto \color{blue}{\sqrt{\left(ew \cdot \sin t\right) \cdot \left(ew \cdot \sin t\right)}} \]

      3. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{ew \cdot \sin t} \cdot \sqrt{ew \cdot \sin t}} \]

      4. rem-square-sqrt 46.1

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

   9. Applied rewrites 46.1%

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

Alternative 3: 88.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ \mathbf{if}\;eh \leq -6 \cdot 10^{-11} \lor \neg \left(eh \leq 3.2 \cdot 10^{+59}\right):\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot t\_1, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} t\_1}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)))
   (if (or (<= eh -6e-11) (not (<= eh 3.2e+59)))
     (fabs (* eh (* (cos t) (sin (atan (/ (* eh (cos t)) (* ew (sin t))))))))
     (fabs (/ (fma (* (cos t) t_1) eh (* (sin t) ew)) (cosh (asinh t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double tmp;
	if ((eh <= -6e-11) || !(eh <= 3.2e+59)) {
		tmp = fabs((eh * (cos(t) * sin(atan(((eh * cos(t)) / (ew * sin(t))))))));
	} else {
		tmp = fabs((fma((cos(t) * t_1), eh, (sin(t) * ew)) / cosh(asinh(t_1))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	tmp = 0.0
	if ((eh <= -6e-11) || !(eh <= 3.2e+59))
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(Float64(eh * cos(t)) / Float64(ew * sin(t))))))));
	else
		tmp = abs(Float64(fma(Float64(cos(t) * t_1), eh, Float64(sin(t) * ew)) / cosh(asinh(t_1))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -6e-11], N[Not[LessEqual[eh, 3.2e+59]], $MachinePrecision]], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * t$95$1), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -6e-11 or 3.19999999999999982e59 < 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. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 87.3

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

   8. Applied rewrites 87.3%

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

   if -6e-11 < eh < 3.19999999999999982e59

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 93.3%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -6 \cdot 10^{-11} \lor \neg \left(eh \leq 3.2 \cdot 10^{+59}\right):\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ \mathbf{if}\;eh \leq -4.6 \cdot 10^{-94} \lor \neg \left(eh \leq 2 \cdot 10^{-64}\right):\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot t\_1, eh, \sin t \cdot ew\right)}{\sqrt{1 + {t\_1}^{2}}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)))
   (if (or (<= eh -4.6e-94) (not (<= eh 2e-64)))
     (fabs (* eh (* (cos t) (sin (atan (/ (* eh (cos t)) (* ew (sin t))))))))
     (fabs
      (/
       (fma (* (cos t) t_1) eh (* (sin t) ew))
       (sqrt (+ 1.0 (pow t_1 2.0))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double tmp;
	if ((eh <= -4.6e-94) || !(eh <= 2e-64)) {
		tmp = fabs((eh * (cos(t) * sin(atan(((eh * cos(t)) / (ew * sin(t))))))));
	} else {
		tmp = fabs((fma((cos(t) * t_1), eh, (sin(t) * ew)) / sqrt((1.0 + pow(t_1, 2.0)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	tmp = 0.0
	if ((eh <= -4.6e-94) || !(eh <= 2e-64))
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(Float64(eh * cos(t)) / Float64(ew * sin(t))))))));
	else
		tmp = abs(Float64(fma(Float64(cos(t) * t_1), eh, Float64(sin(t) * ew)) / sqrt(Float64(1.0 + (t_1 ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -4.6e-94], N[Not[LessEqual[eh, 2e-64]], $MachinePrecision]], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * t$95$1), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.5999999999999999e-94 or 1.99999999999999993e-64 < 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. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 53.0

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 82.3

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

   8. Applied rewrites 82.3%

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

   if -4.5999999999999999e-94 < eh < 1.99999999999999993e-64

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 98.4%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 94.7%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.6 \cdot 10^{-94} \lor \neg \left(eh \leq 2 \cdot 10^{-64}\right):\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2}}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -3 \cdot 10^{-70} \lor \neg \left(eh \leq 2 \cdot 10^{-64}\right):\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -3e-70) (not (<= eh 2e-64)))
   (fabs (* eh (* (cos t) (sin (atan (/ (* eh (cos t)) (* ew (sin t))))))))
   (fabs
    (/
     (fma (/ eh (* ew t)) eh (* (sin t) ew))
     (cosh (asinh (/ (/ eh (tan t)) ew)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3e-70) || !(eh <= 2e-64)) {
		tmp = fabs((eh * (cos(t) * sin(atan(((eh * cos(t)) / (ew * sin(t))))))));
	} else {
		tmp = fabs((fma((eh / (ew * t)), eh, (sin(t) * ew)) / cosh(asinh(((eh / tan(t)) / ew)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -3e-70) || !(eh <= 2e-64))
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(Float64(eh * cos(t)) / Float64(ew * sin(t))))))));
	else
		tmp = abs(Float64(fma(Float64(eh / Float64(ew * t)), eh, Float64(sin(t) * ew)) / cosh(asinh(Float64(Float64(eh / tan(t)) / ew)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -3e-70], N[Not[LessEqual[eh, 2e-64]], $MachinePrecision]], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.0000000000000001e-70 or 1.99999999999999993e-64 < 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. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 54.1

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

   5. Applied rewrites 54.1%

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

   6. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 83.0

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

   8. Applied rewrites 83.0%

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

   if -3.0000000000000001e-70 < eh < 1.99999999999999993e-64

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 84.5

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

   7. Applied rewrites 84.5%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3 \cdot 10^{-70} \lor \neg \left(eh \leq 2 \cdot 10^{-64}\right):\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -2.15 \cdot 10^{+86} \lor \neg \left(eh \leq 1.75 \cdot 10^{+60}\right):\\ \;\;\;\;\left|eh \cdot \tanh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -2.15e+86) (not (<= eh 1.75e+60)))
   (fabs (* eh (tanh (asinh (/ eh (* (tan t) ew))))))
   (fabs
    (/
     (fma (/ eh (* ew t)) eh (* (sin t) ew))
     (cosh (asinh (/ (/ eh (tan t)) ew)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -2.15e+86) || !(eh <= 1.75e+60)) {
		tmp = fabs((eh * tanh(asinh((eh / (tan(t) * ew))))));
	} else {
		tmp = fabs((fma((eh / (ew * t)), eh, (sin(t) * ew)) / cosh(asinh(((eh / tan(t)) / ew)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -2.15e+86) || !(eh <= 1.75e+60))
		tmp = abs(Float64(eh * tanh(asinh(Float64(eh / Float64(tan(t) * ew))))));
	else
		tmp = abs(Float64(fma(Float64(eh / Float64(ew * t)), eh, Float64(sin(t) * ew)) / cosh(asinh(Float64(Float64(eh / tan(t)) / ew)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -2.15e+86], N[Not[LessEqual[eh, 1.75e+60]], $MachinePrecision]], N[Abs[N[(eh * N[Tanh[N[ArcSinh[N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.1500000000000001e86 or 1.7500000000000001e60 < 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. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 55.8

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

   5. Applied rewrites 55.8%

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

   7. Applied rewrites 6.3%

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

   9. Applied rewrites 6.8%

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

   11. Applied rewrites 55.8%

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

   if -2.1500000000000001e86 < eh < 1.7500000000000001e60

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 74.3

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

   7. Applied rewrites 74.3%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.15 \cdot 10^{+86} \lor \neg \left(eh \leq 1.75 \cdot 10^{+60}\right):\\ \;\;\;\;\left|eh \cdot \tanh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -3 \cdot 10^{-70} \lor \neg \left(eh \leq 2 \cdot 10^{-64}\right):\\ \;\;\;\;\left|eh \cdot \tanh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -3e-70) (not (<= eh 2e-64)))
   (fabs (* eh (tanh (asinh (/ eh (* (tan t) ew))))))
   (fabs (* ew (sin t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3e-70) || !(eh <= 2e-64)) {
		tmp = fabs((eh * tanh(asinh((eh / (tan(t) * ew))))));
	} else {
		tmp = fabs((ew * sin(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -3e-70) or not (eh <= 2e-64):
		tmp = math.fabs((eh * math.tanh(math.asinh((eh / (math.tan(t) * ew))))))
	else:
		tmp = math.fabs((ew * math.sin(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -3e-70) || !(eh <= 2e-64))
		tmp = abs(Float64(eh * tanh(asinh(Float64(eh / Float64(tan(t) * ew))))));
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -3e-70) || ~((eh <= 2e-64)))
		tmp = abs((eh * tanh(asinh((eh / (tan(t) * ew))))));
	else
		tmp = abs((ew * sin(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -3e-70], N[Not[LessEqual[eh, 2e-64]], $MachinePrecision]], N[Abs[N[(eh * N[Tanh[N[ArcSinh[N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.0000000000000001e-70 or 1.99999999999999993e-64 < 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. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 54.1

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

   5. Applied rewrites 54.1%

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

   7. Applied rewrites 17.3%

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

   9. Applied rewrites 17.0%

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

   11. Applied rewrites 54.1%

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

   if -3.0000000000000001e-70 < eh < 1.99999999999999993e-64

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 81.1

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

   7. Applied rewrites 81.1%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3 \cdot 10^{-70} \lor \neg \left(eh \leq 2 \cdot 10^{-64}\right):\\ \;\;\;\;\left|eh \cdot \tanh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-124} \lor \neg \left(t \leq 7.8 \cdot 10^{-148}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(-0.5, \frac{eh}{ew}, 0.16666666666666666 \cdot \frac{eh}{ew}\right), \frac{eh}{ew}\right)}{t}\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -1.7e-124) (not (<= t 7.8e-148)))
   (fabs (* ew (sin t)))
   (fabs
    (*
     (sin
      (atan
       (/
        (fma
         (* t t)
         (fma -0.5 (/ eh ew) (* 0.16666666666666666 (/ eh ew)))
         (/ eh ew))
        t)))
     eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -1.7e-124) || !(t <= 7.8e-148)) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs((sin(atan((fma((t * t), fma(-0.5, (eh / ew), (0.16666666666666666 * (eh / ew))), (eh / ew)) / t))) * eh));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -1.7e-124) || !(t <= 7.8e-148))
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(Float64(sin(atan(Float64(fma(Float64(t * t), fma(-0.5, Float64(eh / ew), Float64(0.16666666666666666 * Float64(eh / ew))), Float64(eh / ew)) / t))) * eh));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -1.7e-124], N[Not[LessEqual[t, 7.8e-148]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(t * t), $MachinePrecision] * N[(-0.5 * N[(eh / ew), $MachinePrecision] + N[(0.16666666666666666 * N[(eh / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.7e-124 or 7.79999999999999988e-148 < t

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 51.0

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

   7. Applied rewrites 51.0%

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

   if -1.7e-124 < t < 7.79999999999999988e-148

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 83.3

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

   5. Applied rewrites 83.3%

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

   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 68.5%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-124} \lor \neg \left(t \leq 7.8 \cdot 10^{-148}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(-0.5, \frac{eh}{ew}, 0.16666666666666666 \cdot \frac{eh}{ew}\right), \frac{eh}{ew}\right)}{t}\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.8% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot \sin t\right| \end{array} \]

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((ew * sin(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((ew * sin(t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Sin[t], $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

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-atan.f64 N/A

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

   6. sin-atan N/A

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

   7. associate-*r/ N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

4. Applied rewrites 64.3%

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

5. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 43.5

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

7. Applied rewrites 43.5%

   \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
8. Add Preprocessing

Alternative 10: 18.8% accurate, 108.8× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot t\right| \end{array} \]

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((ew * 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((ew * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-atan.f64 N/A

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

   6. sin-atan N/A

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

   7. associate-*r/ N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

4. Applied rewrites 64.3%

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

5. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 43.5

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

7. Applied rewrites 43.5%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 18.7%

    \[\leadsto \left|ew \cdot \color{blue}{t}\right| \]
11. Add Preprocessing

Alternative 11: 10.1% accurate, 145.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(eh, ew, t):
	return t * ew

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-atan.f64 N/A

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

   6. sin-atan N/A

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

   7. associate-*r/ N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

4. Applied rewrites 64.3%

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

5. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 43.5

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

7. Applied rewrites 43.5%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 18.7%

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

    1. lift-fabs.f64 N/A

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

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

       \[\leadsto \color{blue}{\sqrt{\left(ew \cdot t\right) \cdot \left(ew \cdot t\right)}} \]

    3. sqrt-prod N/A

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

    4. rem-square-sqrt 12.1

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

12. Applied rewrites 12.1%

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

Reproduce

herbie shell --seed 2024337 
(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))))))))