Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 14.7s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 52.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq 2 \cdot 10^{-182}:\\ \;\;\;\;\left|1 \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        2e-182)
     (fabs (* 1.0 ew))
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= 2e-182) {
		tmp = fabs((1.0 * ew));
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan(((eh * tan(t)) / -ew))
    if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= 2d-182) then
        tmp = abs((1.0d0 * ew))
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if ((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))) <= 2e-182) {
		tmp = Math.abs((1.0 * ew));
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if (((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))) <= 2e-182:
		tmp = math.fabs((1.0 * ew))
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= 2e-182)
		tmp = abs(Float64(1.0 * ew));
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= 2e-182)
		tmp = abs((1.0 * ew));
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-182], N[Abs[N[(1.0 * ew), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < 2.0000000000000001e-182

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 44.4%

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

   7. Applied rewrites 44.2%

      \[\leadsto \left|{\left(1 + {\left(\frac{\tan t}{ew} \cdot eh\right)}^{2}\right)}^{-0.5} \cdot ew\right| \]

   8. Taylor expanded in eh around 0

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

   10. Applied rewrites 44.6%

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

   if 2.0000000000000001e-182 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   4. Applied rewrites 63.9%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 59.5

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

   7. Applied rewrites 59.5%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq 2 \cdot 10^{-182}:\\ \;\;\;\;\left|1 \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((((eh * sin(t)) * sin(atan(((-t / ew) * eh)))) - ((ew * cos(t)) * cos(atan(((eh * tan(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 = abs((((eh * sin(t)) * sin(atan(((-t / ew) * eh)))) - ((ew * cos(t)) * cos(atan(((eh * tan(t)) / -ew))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

   5. lower-*.f64 N/A

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

   6. distribute-neg-frac N/A

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

   7. lower-/.f64 N/A

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

   8. lower-neg.f64 99.0

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

5. Applied rewrites 99.0%

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

6. Final simplification 99.0%

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

Alternative 4: 91.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \frac{\tan t}{ew}\\ \mathbf{if}\;eh \leq -8 \cdot 10^{+149} \lor \neg \left(eh \leq 1.25 \cdot 10^{+219}\right):\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t}{ew}, \tanh \left(\frac{eh \cdot t}{ew}\right) \cdot eh, \cos t \cdot \cos \tan^{-1} t\_1\right) \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (/ (tan t) ew))))
   (if (or (<= eh -8e+149) (not (<= eh 1.25e+219)))
     (fabs (* (* (sin t) eh) (tanh (asinh t_1))))
     (fabs
      (*
       (fma
        (/ (sin t) ew)
        (* (tanh (/ (* eh t) ew)) eh)
        (* (cos t) (cos (atan t_1))))
       ew)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * (tan(t) / ew);
	double tmp;
	if ((eh <= -8e+149) || !(eh <= 1.25e+219)) {
		tmp = fabs(((sin(t) * eh) * tanh(asinh(t_1))));
	} else {
		tmp = fabs((fma((sin(t) / ew), (tanh(((eh * t) / ew)) * eh), (cos(t) * cos(atan(t_1)))) * ew));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * Float64(tan(t) / ew))
	tmp = 0.0
	if ((eh <= -8e+149) || !(eh <= 1.25e+219))
		tmp = abs(Float64(Float64(sin(t) * eh) * tanh(asinh(t_1))));
	else
		tmp = abs(Float64(fma(Float64(sin(t) / ew), Float64(tanh(Float64(Float64(eh * t) / ew)) * eh), Float64(cos(t) * cos(atan(t_1)))) * ew));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -8e+149], N[Not[LessEqual[eh, 1.25e+219]], $MachinePrecision]], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * N[(N[Tanh[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * N[Cos[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.00000000000000039e149 or 1.25e219 < eh

   1. Initial program 99.9%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-atan.f64 N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

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

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 86.9%

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

   7. Applied rewrites 86.9%

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

   if -8.00000000000000039e149 < eh < 1.25e219

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 98.3%

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

   7. Applied rewrites 98.3%

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

   9. Applied rewrites 98.3%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 97.2%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -8 \cdot 10^{+149} \lor \neg \left(eh \leq 1.25 \cdot 10^{+219}\right):\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t}{ew}, \tanh \left(\frac{eh \cdot t}{ew}\right) \cdot eh, \cos t \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right) \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -3.3 \cdot 10^{+96} \lor \neg \left(eh \leq 1.6 \cdot 10^{+141}\right):\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -3.3e+96) (not (<= eh 1.6e+141)))
   (fabs (* (* (sin t) eh) (tanh (asinh (* eh (/ (tan t) ew))))))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3.3e+96) || !(eh <= 1.6e+141)) {
		tmp = fabs(((sin(t) * eh) * tanh(asinh((eh * (tan(t) / ew))))));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -3.3e+96) or not (eh <= 1.6e+141):
		tmp = math.fabs(((math.sin(t) * eh) * math.tanh(math.asinh((eh * (math.tan(t) / ew))))))
	else:
		tmp = math.fabs((ew * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -3.3e+96) || !(eh <= 1.6e+141))
		tmp = abs(Float64(Float64(sin(t) * eh) * tanh(asinh(Float64(eh * Float64(tan(t) / ew))))));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -3.3e+96) || ~((eh <= 1.6e+141)))
		tmp = abs(((sin(t) * eh) * tanh(asinh((eh * (tan(t) / ew))))));
	else
		tmp = abs((ew * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -3.3e+96], N[Not[LessEqual[eh, 1.6e+141]], $MachinePrecision]], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.29999999999999984e96 or 1.60000000000000009e141 < eh

   1. Initial program 99.9%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-atan.f64 N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

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

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 78.6%

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

   7. Applied rewrites 78.6%

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

   if -3.29999999999999984e96 < eh < 1.60000000000000009e141

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.2%

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in eh around 0

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

       1. lower-*.f64 N/A

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

       2. lower-cos.f64 78.6

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

   12. Applied rewrites 78.6%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.3 \cdot 10^{+96} \lor \neg \left(eh \leq 1.6 \cdot 10^{+141}\right):\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -3.3 \cdot 10^{+96} \lor \neg \left(eh \leq 4.4 \cdot 10^{+221}\right):\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(\frac{eh}{ew} \cdot 0.3333333333333333, t \cdot t, \frac{eh}{ew}\right) \cdot \left(-t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -3.3e+96) (not (<= eh 4.4e+221)))
   (fabs
    (*
     (* (- eh) (sin t))
     (sin
      (atan
       (* (fma (* (/ eh ew) 0.3333333333333333) (* t t) (/ eh ew)) (- t))))))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3.3e+96) || !(eh <= 4.4e+221)) {
		tmp = fabs(((-eh * sin(t)) * sin(atan((fma(((eh / ew) * 0.3333333333333333), (t * t), (eh / ew)) * -t)))));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -3.3e+96) || !(eh <= 4.4e+221))
		tmp = abs(Float64(Float64(Float64(-eh) * sin(t)) * sin(atan(Float64(fma(Float64(Float64(eh / ew) * 0.3333333333333333), Float64(t * t), Float64(eh / ew)) * Float64(-t))))));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -3.3e+96], N[Not[LessEqual[eh, 4.4e+221]], $MachinePrecision]], N[Abs[N[(N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[(N[(eh / ew), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] * (-t)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.29999999999999984e96 or 4.3999999999999999e221 < eh

   1. Initial program 99.9%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-atan.f64 N/A

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

      11. mul-1-neg N/A

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

      12. *-commutative N/A

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

      13. times-frac N/A

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

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

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 77.3%

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

   if -3.29999999999999984e96 < eh < 4.3999999999999999e221

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 97.7%

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

   7. Applied rewrites 97.7%

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

   9. Applied rewrites 97.7%

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

   10. Taylor expanded in eh around 0

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

       1. lower-*.f64 N/A

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

       2. lower-cos.f64 75.1

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

   12. Applied rewrites 75.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.3 \cdot 10^{+96} \lor \neg \left(eh \leq 4.4 \cdot 10^{+221}\right):\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(\frac{eh}{ew} \cdot 0.3333333333333333, t \cdot t, \frac{eh}{ew}\right) \cdot \left(-t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.0% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in ew around inf

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

5. Applied rewrites 91.3%

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

7. Applied rewrites 91.3%

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

9. Applied rewrites 91.3%

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

10. Taylor expanded in eh around 0

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

    1. lower-*.f64 N/A

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

    2. lower-cos.f64 61.8

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

12. Applied rewrites 61.8%

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

Alternative 8: 42.1% accurate, 107.8× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((1.0 * 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 = abs((1.0d0 * ew))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 42.7%

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

7. Applied rewrites 42.3%

   \[\leadsto \left|{\left(1 + {\left(\frac{\tan t}{ew} \cdot eh\right)}^{2}\right)}^{-0.5} \cdot ew\right| \]

8. Taylor expanded in eh around 0

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

10. Applied rewrites 42.8%

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

Reproduce

herbie shell --seed 2024305 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))