Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.2s

Alternatives: 11

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

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

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(eh * tan(t)) / Float64(-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]]

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%

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

Alternative 2: 52.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -1 \cdot 10^{-281}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew}\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<= (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))) -1e-281)
     (/ 1.0 (fabs (/ 1.0 ew)))
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -1e-281) {
		tmp = 1.0 / fabs((1.0 / ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * cos(t)
    t_2 = atan(((eh * tan(t)) / -ew))
    if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= (-1d-281)) then
        tmp = 1.0d0 / abs((1.0d0 / ew))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if (((t_1 * Math.cos(t_2)) - ((eh * Math.sin(t)) * Math.sin(t_2))) <= -1e-281) {
		tmp = 1.0 / Math.abs((1.0 / ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if ((t_1 * math.cos(t_2)) - ((eh * math.sin(t)) * math.sin(t_2))) <= -1e-281:
		tmp = 1.0 / math.fabs((1.0 / ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= -1e-281)
		tmp = Float64(1.0 / abs(Float64(1.0 / ew)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -1e-281)
		tmp = 1.0 / abs((1.0 / ew));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-281], N[(1.0 / N[Abs[N[(1.0 / ew), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

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))))) < -1e-281

   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 80.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 43.9

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

   7. Applied rewrites 43.9%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

   if -1e-281 < (-.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 75.3%

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

   6. Applied rewrites 59.2%

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

   7. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 58.0

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

   9. Applied rewrites 58.0%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -1 \cdot 10^{-281}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew}\right|}\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh \cdot \tan t}{ew}\\ t_2 := \sin t \cdot \sin \tan^{-1} t\_1\\ t_3 := \left|eh \cdot \mathsf{fma}\left(\frac{1}{eh \cdot \sqrt{1 + {t\_1}^{2}}}, ew \cdot \cos t, t\_2\right)\right|\\ \mathbf{if}\;eh \leq -3.7 \cdot 10^{+102}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eh \leq 7.5 \cdot 10^{-43}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{t\_2}{ew}, \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (* eh (tan t)) ew))
        (t_2 (* (sin t) (sin (atan t_1))))
        (t_3
         (fabs
          (*
           eh
           (fma
            (/ 1.0 (* eh (sqrt (+ 1.0 (pow t_1 2.0)))))
            (* ew (cos t))
            t_2)))))
   (if (<= eh -3.7e+102)
     t_3
     (if (<= eh 7.5e-43) (fabs (* ew (fma eh (/ t_2 ew) (cos t)))) t_3))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh * tan(t)) / ew;
	double t_2 = sin(t) * sin(atan(t_1));
	double t_3 = fabs((eh * fma((1.0 / (eh * sqrt((1.0 + pow(t_1, 2.0))))), (ew * cos(t)), t_2)));
	double tmp;
	if (eh <= -3.7e+102) {
		tmp = t_3;
	} else if (eh <= 7.5e-43) {
		tmp = fabs((ew * fma(eh, (t_2 / ew), cos(t))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh * tan(t)) / ew)
	t_2 = Float64(sin(t) * sin(atan(t_1)))
	t_3 = abs(Float64(eh * fma(Float64(1.0 / Float64(eh * sqrt(Float64(1.0 + (t_1 ^ 2.0))))), Float64(ew * cos(t)), t_2)))
	tmp = 0.0
	if (eh <= -3.7e+102)
		tmp = t_3;
	elseif (eh <= 7.5e-43)
		tmp = abs(Float64(ew * fma(eh, Float64(t_2 / ew), cos(t))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(eh * N[(N[(1.0 / N[(eh * N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -3.7e+102], t$95$3, If[LessEqual[eh, 7.5e-43], N[Abs[N[(ew * N[(eh * N[(t$95$2 / ew), $MachinePrecision] + N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.70000000000000023e102 or 7.50000000000000068e-43 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-tan.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 31.3

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

   5. Applied rewrites 31.3%

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

   6. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   8. Applied rewrites 99.8%

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

   10. Applied rewrites 99.7%

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

   if -3.70000000000000023e102 < eh < 7.50000000000000068e-43

   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

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-tan.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 48.6

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

   5. Applied rewrites 48.6%

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

   6. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   8. Applied rewrites 81.7%

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

   10. Applied rewrites 81.5%

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

   11. Taylor expanded in ew around inf

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

   13. Applied rewrites 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.7 \cdot 10^{+102}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(\frac{1}{eh \cdot \sqrt{1 + {\left(\frac{eh \cdot \tan t}{ew}\right)}^{2}}}, ew \cdot \cos t, \sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\right)\right|\\ \mathbf{elif}\;eh \leq 7.5 \cdot 10^{-43}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)}{ew}, \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(\frac{1}{eh \cdot \sqrt{1 + {\left(\frac{eh \cdot \tan t}{ew}\right)}^{2}}}, ew \cdot \cos t, \sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\\ t_2 := \left|eh \cdot \mathsf{fma}\left(\frac{1}{eh}, ew \cdot \cos t, t\_1\right)\right|\\ \mathbf{if}\;eh \leq -4.5 \cdot 10^{+102}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 10^{+38}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{t\_1}{ew}, \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) (sin (atan (/ (* eh (tan t)) ew)))))
        (t_2 (fabs (* eh (fma (/ 1.0 eh) (* ew (cos t)) t_1)))))
   (if (<= eh -4.5e+102)
     t_2
     (if (<= eh 1e+38) (fabs (* ew (fma eh (/ t_1 ew) (cos t)))) t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * sin(atan(((eh * tan(t)) / ew)));
	double t_2 = fabs((eh * fma((1.0 / eh), (ew * cos(t)), t_1)));
	double tmp;
	if (eh <= -4.5e+102) {
		tmp = t_2;
	} else if (eh <= 1e+38) {
		tmp = fabs((ew * fma(eh, (t_1 / ew), cos(t))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * sin(atan(Float64(Float64(eh * tan(t)) / ew))))
	t_2 = abs(Float64(eh * fma(Float64(1.0 / eh), Float64(ew * cos(t)), t_1)))
	tmp = 0.0
	if (eh <= -4.5e+102)
		tmp = t_2;
	elseif (eh <= 1e+38)
		tmp = abs(Float64(ew * fma(eh, Float64(t_1 / ew), cos(t))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(eh * N[(N[(1.0 / eh), $MachinePrecision] * N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -4.5e+102], t$95$2, If[LessEqual[eh, 1e+38], N[Abs[N[(ew * N[(eh * N[(t$95$1 / ew), $MachinePrecision] + N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.50000000000000021e102 or 9.99999999999999977e37 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-tan.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 27.4

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

   5. Applied rewrites 27.4%

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

   6. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   8. Applied rewrites 99.8%

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

   10. Applied rewrites 99.7%

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

   11. Taylor expanded in eh around 0

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

   13. Applied rewrites 98.6%

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

   if -4.50000000000000021e102 < eh < 9.99999999999999977e37

   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

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-tan.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   8. Applied rewrites 83.7%

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

   10. Applied rewrites 83.5%

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

   11. Taylor expanded in ew around inf

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

   13. Applied rewrites 98.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.5 \cdot 10^{+102}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(\frac{1}{eh}, ew \cdot \cos t, \sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\right)\right|\\ \mathbf{elif}\;eh \leq 10^{+38}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)}{ew}, \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(\frac{1}{eh}, ew \cdot \cos t, \sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;eh \leq -5.8 \cdot 10^{+200}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 1.3 \cdot 10^{+178}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)}{ew}, \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= eh -5.8e+200)
     t_1
     (if (<= eh 1.3e+178)
       (fabs
        (*
         ew
         (fma
          eh
          (/ (* (sin t) (sin (atan (/ (* eh (tan t)) ew)))) ew)
          (cos t))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (eh <= -5.8e+200) {
		tmp = t_1;
	} else if (eh <= 1.3e+178) {
		tmp = fabs((ew * fma(eh, ((sin(t) * sin(atan(((eh * tan(t)) / ew)))) / ew), cos(t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (eh <= -5.8e+200)
		tmp = t_1;
	elseif (eh <= 1.3e+178)
		tmp = abs(Float64(ew * fma(eh, Float64(Float64(sin(t) * sin(atan(Float64(Float64(eh * tan(t)) / ew)))) / ew), cos(t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -5.8e+200], t$95$1, If[LessEqual[eh, 1.3e+178], N[Abs[N[(ew * N[(eh * N[(N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.7999999999999998e200 or 1.3e178 < eh

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 87.0

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

   7. Applied rewrites 87.0%

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

   if -5.7999999999999998e200 < eh < 1.3e178

   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

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-tan.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 46.9

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   8. Applied rewrites 87.5%

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

   10. Applied rewrites 87.3%

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

   11. Taylor expanded in ew around inf

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

   13. Applied rewrites 96.9%

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

Alternative 6: 75.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;eh \leq -7.5 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 2.65 \cdot 10^{+39}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= eh -7.5e+79) t_1 (if (<= eh 2.65e+39) (fabs (* ew (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (eh <= -7.5e+79) {
		tmp = t_1;
	} else if (eh <= 2.65e+39) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((eh * sin(t)))
    if (eh <= (-7.5d+79)) then
        tmp = t_1
    else if (eh <= 2.65d+39) then
        tmp = abs((ew * cos(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * Math.sin(t)));
	double tmp;
	if (eh <= -7.5e+79) {
		tmp = t_1;
	} else if (eh <= 2.65e+39) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	tmp = 0
	if eh <= -7.5e+79:
		tmp = t_1
	elif eh <= 2.65e+39:
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (eh <= -7.5e+79)
		tmp = t_1;
	elseif (eh <= 2.65e+39)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	tmp = 0.0;
	if (eh <= -7.5e+79)
		tmp = t_1;
	elseif (eh <= 2.65e+39)
		tmp = abs((ew * cos(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -7.5e+79], t$95$1, If[LessEqual[eh, 2.65e+39], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -7.49999999999999967e79 or 2.64999999999999989e39 < eh

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 51.8%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 74.5

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

   7. Applied rewrites 74.5%

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

   if -7.49999999999999967e79 < eh < 2.64999999999999989e39

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 83.7

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

   7. Applied rewrites 83.7%

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

Alternative 7: 61.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew}\right|}\\ \mathbf{elif}\;ew \leq 5.2 \cdot 10^{-27}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.8e-126)
   (/ 1.0 (fabs (/ 1.0 ew)))
   (if (<= ew 5.2e-27) (fabs (* eh (sin t))) (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.8e-126) {
		tmp = 1.0 / fabs((1.0 / ew));
	} else if (ew <= 5.2e-27) {
		tmp = fabs((eh * sin(t)));
	} else {
		tmp = ew * cos(t);
	}
	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) :: tmp
    if (ew <= (-1.8d-126)) then
        tmp = 1.0d0 / abs((1.0d0 / ew))
    else if (ew <= 5.2d-27) then
        tmp = abs((eh * sin(t)))
    else
        tmp = ew * cos(t)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.8e-126) {
		tmp = 1.0 / Math.abs((1.0 / ew));
	} else if (ew <= 5.2e-27) {
		tmp = Math.abs((eh * Math.sin(t)));
	} else {
		tmp = ew * Math.cos(t);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1.8e-126:
		tmp = 1.0 / math.fabs((1.0 / ew))
	elif ew <= 5.2e-27:
		tmp = math.fabs((eh * math.sin(t)))
	else:
		tmp = ew * math.cos(t)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.8e-126)
		tmp = Float64(1.0 / abs(Float64(1.0 / ew)));
	elseif (ew <= 5.2e-27)
		tmp = abs(Float64(eh * sin(t)));
	else
		tmp = Float64(ew * cos(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1.8e-126)
		tmp = 1.0 / abs((1.0 / ew));
	elseif (ew <= 5.2e-27)
		tmp = abs((eh * sin(t)));
	else
		tmp = ew * cos(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.8e-126], N[(1.0 / N[Abs[N[(1.0 / ew), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[ew, 5.2e-27], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.8e-126

   1. Initial program 99.7%

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 50.0

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

   7. Applied rewrites 50.0%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

   if -1.8e-126 < ew < 5.20000000000000034e-27

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 53.6%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 76.1

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

   7. Applied rewrites 76.1%

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

   if 5.20000000000000034e-27 < 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 92.5%

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

   6. Applied rewrites 48.3%

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

   7. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 59.8

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

   9. Applied rewrites 59.8%

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

Alternative 8: 43.8% accurate, 24.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.6 \cdot 10^{-127}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{ew}\right|}\\ \mathbf{elif}\;ew \leq 3.8 \cdot 10^{-34}:\\ \;\;\;\;\left|t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{ew}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.6e-127)
   (/ 1.0 (fabs (/ 1.0 ew)))
   (if (<= ew 3.8e-34) (fabs (* t eh)) (/ 1.0 (/ 1.0 ew)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.6e-127) {
		tmp = 1.0 / fabs((1.0 / ew));
	} else if (ew <= 3.8e-34) {
		tmp = fabs((t * eh));
	} else {
		tmp = 1.0 / (1.0 / 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) :: tmp
    if (ew <= (-1.6d-127)) then
        tmp = 1.0d0 / abs((1.0d0 / ew))
    else if (ew <= 3.8d-34) then
        tmp = abs((t * eh))
    else
        tmp = 1.0d0 / (1.0d0 / ew)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.6e-127) {
		tmp = 1.0 / Math.abs((1.0 / ew));
	} else if (ew <= 3.8e-34) {
		tmp = Math.abs((t * eh));
	} else {
		tmp = 1.0 / (1.0 / ew);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1.6e-127:
		tmp = 1.0 / math.fabs((1.0 / ew))
	elif ew <= 3.8e-34:
		tmp = math.fabs((t * eh))
	else:
		tmp = 1.0 / (1.0 / ew)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.6e-127)
		tmp = Float64(1.0 / abs(Float64(1.0 / ew)));
	elseif (ew <= 3.8e-34)
		tmp = abs(Float64(t * eh));
	else
		tmp = Float64(1.0 / Float64(1.0 / ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1.6e-127)
		tmp = 1.0 / abs((1.0 / ew));
	elseif (ew <= 3.8e-34)
		tmp = abs((t * eh));
	else
		tmp = 1.0 / (1.0 / ew);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.6e-127], N[(1.0 / N[Abs[N[(1.0 / ew), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[ew, 3.8e-34], N[Abs[N[(t * eh), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(1.0 / ew), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.60000000000000009e-127

   1. Initial program 99.7%

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 50.0

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

   7. Applied rewrites 50.0%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

   if -1.60000000000000009e-127 < ew < 3.8000000000000001e-34

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 53.1%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 75.9

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 44.8%

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

   if 3.8000000000000001e-34 < 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 92.6%

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in t around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{ew}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 48.6

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{ew}}} \]

   9. Applied rewrites 48.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{ew}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 40.7% accurate, 24.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.65 \cdot 10^{-127}:\\ \;\;\;\;\left|eh \cdot \frac{ew}{eh}\right|\\ \mathbf{elif}\;ew \leq 3.8 \cdot 10^{-34}:\\ \;\;\;\;\left|t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{ew}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.65e-127)
   (fabs (* eh (/ ew eh)))
   (if (<= ew 3.8e-34) (fabs (* t eh)) (/ 1.0 (/ 1.0 ew)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.65e-127) {
		tmp = fabs((eh * (ew / eh)));
	} else if (ew <= 3.8e-34) {
		tmp = fabs((t * eh));
	} else {
		tmp = 1.0 / (1.0 / 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) :: tmp
    if (ew <= (-1.65d-127)) then
        tmp = abs((eh * (ew / eh)))
    else if (ew <= 3.8d-34) then
        tmp = abs((t * eh))
    else
        tmp = 1.0d0 / (1.0d0 / ew)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.65e-127) {
		tmp = Math.abs((eh * (ew / eh)));
	} else if (ew <= 3.8e-34) {
		tmp = Math.abs((t * eh));
	} else {
		tmp = 1.0 / (1.0 / ew);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1.65e-127:
		tmp = math.fabs((eh * (ew / eh)))
	elif ew <= 3.8e-34:
		tmp = math.fabs((t * eh))
	else:
		tmp = 1.0 / (1.0 / ew)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.65e-127)
		tmp = abs(Float64(eh * Float64(ew / eh)));
	elseif (ew <= 3.8e-34)
		tmp = abs(Float64(t * eh));
	else
		tmp = Float64(1.0 / Float64(1.0 / ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1.65e-127)
		tmp = abs((eh * (ew / eh)));
	elseif (ew <= 3.8e-34)
		tmp = abs((t * eh));
	else
		tmp = 1.0 / (1.0 / ew);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.65e-127], N[Abs[N[(eh * N[(ew / eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 3.8e-34], N[Abs[N[(t * eh), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(1.0 / ew), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.6499999999999999e-127

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-tan.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 50.0

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

   5. Applied rewrites 50.0%

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

   6. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   8. Applied rewrites 85.3%

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

   10. Applied rewrites 85.1%

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

   11. Taylor expanded in t around 0

       \[\leadsto \left|eh \cdot \frac{ew}{\color{blue}{eh}}\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 43.7%

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

   if -1.6499999999999999e-127 < ew < 3.8000000000000001e-34

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 53.1%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 75.9

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 44.8%

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

   if 3.8000000000000001e-34 < 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 92.6%

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

   6. Applied rewrites 49.1%

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

   7. Taylor expanded in t around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{ew}}} \]
   8. Step-by-step derivation

      1. lower-/.f64 48.6

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{ew}}} \]

   9. Applied rewrites 48.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{ew}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 37.4% accurate, 27.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \frac{ew}{eh}\right|\\ \mathbf{if}\;ew \leq -1.65 \cdot 10^{-127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 3.8 \cdot 10^{-34}:\\ \;\;\;\;\left|t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (/ ew eh)))))
   (if (<= ew -1.65e-127) t_1 (if (<= ew 3.8e-34) (fabs (* t eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * (ew / eh)));
	double tmp;
	if (ew <= -1.65e-127) {
		tmp = t_1;
	} else if (ew <= 3.8e-34) {
		tmp = fabs((t * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((eh * (ew / eh)))
    if (ew <= (-1.65d-127)) then
        tmp = t_1
    else if (ew <= 3.8d-34) then
        tmp = abs((t * eh))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * (ew / eh)));
	double tmp;
	if (ew <= -1.65e-127) {
		tmp = t_1;
	} else if (ew <= 3.8e-34) {
		tmp = Math.abs((t * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * (ew / eh)))
	tmp = 0
	if ew <= -1.65e-127:
		tmp = t_1
	elif ew <= 3.8e-34:
		tmp = math.fabs((t * eh))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * Float64(ew / eh)))
	tmp = 0.0
	if (ew <= -1.65e-127)
		tmp = t_1;
	elseif (ew <= 3.8e-34)
		tmp = abs(Float64(t * eh));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * (ew / eh)));
	tmp = 0.0;
	if (ew <= -1.65e-127)
		tmp = t_1;
	elseif (ew <= 3.8e-34)
		tmp = abs((t * eh));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[(ew / eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.65e-127], t$95$1, If[LessEqual[ew, 3.8e-34], N[Abs[N[(t * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.6499999999999999e-127 or 3.8000000000000001e-34 < 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

   3. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-tan.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   8. Applied rewrites 84.3%

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

   10. Applied rewrites 84.1%

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

   11. Taylor expanded in t around 0

       \[\leadsto \left|eh \cdot \frac{ew}{\color{blue}{eh}}\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 42.6%

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

   if -1.6499999999999999e-127 < ew < 3.8000000000000001e-34

   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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. un-div-inv N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

   4. Applied rewrites 53.1%

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

   5. Taylor expanded in ew around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 75.9

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 44.8%

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

Alternative 11: 18.6% accurate, 107.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(t * eh), $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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-atan.f64 N/A

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

   5. cos-atan N/A

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

   6. un-div-inv N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

4. Applied rewrites 78.0%

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

5. Taylor expanded in ew around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 42.3

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

7. Applied rewrites 42.3%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 23.2%

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

Reproduce

herbie shell --seed 2024237 
(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)))))))