Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.9% → 99.8%

Time: 9.2s

Alternatives: 10

Speedup: 1.0×

• 49.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\sin x \cdot \sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (sin x) (sinh y)) x))

C

double code(double x, double y) {
	return (sin(x) * sinh(y)) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (sin(x) * sinh(y)) / x
end function

Java

public static double code(double x, double y) {
	return (Math.sin(x) * Math.sinh(y)) / x;
}

Python

def code(x, y):
	return (math.sin(x) * math.sinh(y)) / x

Julia

function code(x, y)
	return Float64(Float64(sin(x) * sinh(y)) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = (sin(x) * sinh(y)) / x;
end

Wolfram

code[x_, y_] := N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Initial Program: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin x \cdot \sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (sin x) (sinh y)) x))

C

double code(double x, double y) {
	return (sin(x) * sinh(y)) / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (sin(x) * sinh(y)) / x
end function

Java

public static double code(double x, double y) {
	return (Math.sin(x) * Math.sinh(y)) / x;
}

Python

def code(x, y):
	return (math.sin(x) * math.sinh(y)) / x

Julia

function code(x, y)
	return Float64(Float64(sin(x) * sinh(y)) / x)
end

MATLAB

function tmp = code(x, y)
	tmp = (sin(x) * sinh(y)) / x;
end

Wolfram

code[x_, y_] := N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{\sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) x)))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / x)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / x);
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / x)

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / x);
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.3%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Step-by-step derivation

   1. associate-/l* 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing

5. Final simplification 99.9%

   \[\leadsto \sin x \cdot \frac{\sinh y}{x} \]
6. Add Preprocessing

Alternative 2: 67.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\\ t_1 := \mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)\\ \mathbf{if}\;y \leq 6400:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+180}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y + 0.16666666666666666 \cdot {y}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (* -0.16666666666666666 (* y (pow x 2.0)))))
        (t_1 (log1p (expm1 y))))
   (if (<= y 6400.0)
     (/ y (/ x (sin x)))
     (if (<= y 4.8e+40)
       t_1
       (if (<= y 5.1e+46)
         t_0
         (if (<= y 1.35e+155)
           t_1
           (if (<= y 4.1e+180)
             t_0
             (+ y (* 0.16666666666666666 (pow y 3.0))))))))))

C

double code(double x, double y) {
	double t_0 = y + (-0.16666666666666666 * (y * pow(x, 2.0)));
	double t_1 = log1p(expm1(y));
	double tmp;
	if (y <= 6400.0) {
		tmp = y / (x / sin(x));
	} else if (y <= 4.8e+40) {
		tmp = t_1;
	} else if (y <= 5.1e+46) {
		tmp = t_0;
	} else if (y <= 1.35e+155) {
		tmp = t_1;
	} else if (y <= 4.1e+180) {
		tmp = t_0;
	} else {
		tmp = y + (0.16666666666666666 * pow(y, 3.0));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = y + (-0.16666666666666666 * (y * Math.pow(x, 2.0)));
	double t_1 = Math.log1p(Math.expm1(y));
	double tmp;
	if (y <= 6400.0) {
		tmp = y / (x / Math.sin(x));
	} else if (y <= 4.8e+40) {
		tmp = t_1;
	} else if (y <= 5.1e+46) {
		tmp = t_0;
	} else if (y <= 1.35e+155) {
		tmp = t_1;
	} else if (y <= 4.1e+180) {
		tmp = t_0;
	} else {
		tmp = y + (0.16666666666666666 * Math.pow(y, 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (-0.16666666666666666 * (y * math.pow(x, 2.0)))
	t_1 = math.log1p(math.expm1(y))
	tmp = 0
	if y <= 6400.0:
		tmp = y / (x / math.sin(x))
	elif y <= 4.8e+40:
		tmp = t_1
	elif y <= 5.1e+46:
		tmp = t_0
	elif y <= 1.35e+155:
		tmp = t_1
	elif y <= 4.1e+180:
		tmp = t_0
	else:
		tmp = y + (0.16666666666666666 * math.pow(y, 3.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(-0.16666666666666666 * Float64(y * (x ^ 2.0))))
	t_1 = log1p(expm1(y))
	tmp = 0.0
	if (y <= 6400.0)
		tmp = Float64(y / Float64(x / sin(x)));
	elseif (y <= 4.8e+40)
		tmp = t_1;
	elseif (y <= 5.1e+46)
		tmp = t_0;
	elseif (y <= 1.35e+155)
		tmp = t_1;
	elseif (y <= 4.1e+180)
		tmp = t_0;
	else
		tmp = Float64(y + Float64(0.16666666666666666 * (y ^ 3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(-0.16666666666666666 * N[(y * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[1 + N[(Exp[y] - 1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, 6400.0], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+40], t$95$1, If[LessEqual[y, 5.1e+46], t$95$0, If[LessEqual[y, 1.35e+155], t$95$1, If[LessEqual[y, 4.1e+180], t$95$0, N[(y + N[(0.16666666666666666 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 6400

   1. Initial program 83.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 49.8%

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   6. Step-by-step derivation

      1. associate-/l* 66.5%

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   7. Simplified 66.5%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]
   8. Step-by-step derivation

      1. clear-num 66.5%

         \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{x}{\sin x}}} \]

      2. un-div-inv 66.5%

         \[\leadsto \color{blue}{\frac{y}{\frac{x}{\sin x}}} \]

   9. Applied egg-rr 66.5%

      \[\leadsto \color{blue}{\frac{y}{\frac{x}{\sin x}}} \]

   if 6400 < y < 4.8e40 or 5.0999999999999997e46 < y < 1.34999999999999997e155

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 3.4%

      \[\leadsto \frac{\sin x \cdot \color{blue}{y}}{x} \]

   4. Taylor expanded in x around 0 6.9%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{x} \]
   5. Step-by-step derivation

      1. *-commutative 6.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{x} \]

   6. Simplified 6.9%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{x} \]
   7. Step-by-step derivation

      1. associate-/l* 3.2%

         \[\leadsto \color{blue}{y \cdot \frac{x}{x}} \]

      2. *-inverses 3.2%

         \[\leadsto y \cdot \color{blue}{1} \]

      3. *-commutative 3.2%

         \[\leadsto \color{blue}{1 \cdot y} \]

      4. *-un-lft-identity 3.2%

         \[\leadsto \color{blue}{y} \]

      5. log1p-expm1-u 80.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)} \]

   8. Applied egg-rr 80.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)} \]

   if 4.8e40 < y < 5.0999999999999997e46 or 1.34999999999999997e155 < y < 4.1e180

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 2.9%

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   6. Step-by-step derivation

      1. associate-/l* 2.9%

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   7. Simplified 2.9%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   8. Taylor expanded in x around 0 88.2%

      \[\leadsto \color{blue}{y + -0.16666666666666666 \cdot \left({x}^{2} \cdot y\right)} \]

   if 4.1e180 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 100.0%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y + 0.16666666666666666 \cdot {y}^{3}\right)}}{x} \]

   4. Taylor expanded in x around 0 80.0%

      \[\leadsto \color{blue}{y + 0.16666666666666666 \cdot {y}^{3}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6400:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+46}:\\ \;\;\;\;y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+155}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+180}:\\ \;\;\;\;y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;y + 0.16666666666666666 \cdot {y}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 \cdot {y}^{3}\\ t_1 := y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\\ \mathbf{if}\;y \leq 620:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+155}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+180}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;y + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.16666666666666666 (pow y 3.0)))
        (t_1 (+ y (* -0.16666666666666666 (* y (pow x 2.0))))))
   (if (<= y 620.0)
     (/ y (/ x (sin x)))
     (if (<= y 3.3e+104)
       t_1
       (if (<= y 1.35e+155) t_0 (if (<= y 4.1e+180) t_1 (+ y t_0)))))))

C

double code(double x, double y) {
	double t_0 = 0.16666666666666666 * pow(y, 3.0);
	double t_1 = y + (-0.16666666666666666 * (y * pow(x, 2.0)));
	double tmp;
	if (y <= 620.0) {
		tmp = y / (x / sin(x));
	} else if (y <= 3.3e+104) {
		tmp = t_1;
	} else if (y <= 1.35e+155) {
		tmp = t_0;
	} else if (y <= 4.1e+180) {
		tmp = t_1;
	} else {
		tmp = y + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.16666666666666666d0 * (y ** 3.0d0)
    t_1 = y + ((-0.16666666666666666d0) * (y * (x ** 2.0d0)))
    if (y <= 620.0d0) then
        tmp = y / (x / sin(x))
    else if (y <= 3.3d+104) then
        tmp = t_1
    else if (y <= 1.35d+155) then
        tmp = t_0
    else if (y <= 4.1d+180) then
        tmp = t_1
    else
        tmp = y + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.16666666666666666 * Math.pow(y, 3.0);
	double t_1 = y + (-0.16666666666666666 * (y * Math.pow(x, 2.0)));
	double tmp;
	if (y <= 620.0) {
		tmp = y / (x / Math.sin(x));
	} else if (y <= 3.3e+104) {
		tmp = t_1;
	} else if (y <= 1.35e+155) {
		tmp = t_0;
	} else if (y <= 4.1e+180) {
		tmp = t_1;
	} else {
		tmp = y + t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.16666666666666666 * math.pow(y, 3.0)
	t_1 = y + (-0.16666666666666666 * (y * math.pow(x, 2.0)))
	tmp = 0
	if y <= 620.0:
		tmp = y / (x / math.sin(x))
	elif y <= 3.3e+104:
		tmp = t_1
	elif y <= 1.35e+155:
		tmp = t_0
	elif y <= 4.1e+180:
		tmp = t_1
	else:
		tmp = y + t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.16666666666666666 * (y ^ 3.0))
	t_1 = Float64(y + Float64(-0.16666666666666666 * Float64(y * (x ^ 2.0))))
	tmp = 0.0
	if (y <= 620.0)
		tmp = Float64(y / Float64(x / sin(x)));
	elseif (y <= 3.3e+104)
		tmp = t_1;
	elseif (y <= 1.35e+155)
		tmp = t_0;
	elseif (y <= 4.1e+180)
		tmp = t_1;
	else
		tmp = Float64(y + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.16666666666666666 * (y ^ 3.0);
	t_1 = y + (-0.16666666666666666 * (y * (x ^ 2.0)));
	tmp = 0.0;
	if (y <= 620.0)
		tmp = y / (x / sin(x));
	elseif (y <= 3.3e+104)
		tmp = t_1;
	elseif (y <= 1.35e+155)
		tmp = t_0;
	elseif (y <= 4.1e+180)
		tmp = t_1;
	else
		tmp = y + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.16666666666666666 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y + N[(-0.16666666666666666 * N[(y * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 620.0], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.3e+104], t$95$1, If[LessEqual[y, 1.35e+155], t$95$0, If[LessEqual[y, 4.1e+180], t$95$1, N[(y + t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 620

   1. Initial program 83.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 49.8%

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   6. Step-by-step derivation

      1. associate-/l* 66.5%

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   7. Simplified 66.5%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]
   8. Step-by-step derivation

      1. clear-num 66.5%

         \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{x}{\sin x}}} \]

      2. un-div-inv 66.5%

         \[\leadsto \color{blue}{\frac{y}{\frac{x}{\sin x}}} \]

   9. Applied egg-rr 66.5%

      \[\leadsto \color{blue}{\frac{y}{\frac{x}{\sin x}}} \]

   if 620 < y < 3.29999999999999985e104 or 1.34999999999999997e155 < y < 4.1e180

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 3.0%

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   6. Step-by-step derivation

      1. associate-/l* 3.0%

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   7. Simplified 3.0%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   8. Taylor expanded in x around 0 46.5%

      \[\leadsto \color{blue}{y + -0.16666666666666666 \cdot \left({x}^{2} \cdot y\right)} \]

   if 3.29999999999999985e104 < y < 1.34999999999999997e155

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 100.0%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y + 0.16666666666666666 \cdot {y}^{3}\right)}}{x} \]

   4. Taylor expanded in x around 0 76.9%

      \[\leadsto \frac{\color{blue}{x \cdot \left(y + 0.16666666666666666 \cdot {y}^{3}\right)}}{x} \]

   5. Taylor expanded in y around inf 76.9%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot {y}^{3}} \]

   if 4.1e180 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 100.0%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y + 0.16666666666666666 \cdot {y}^{3}\right)}}{x} \]

   4. Taylor expanded in x around 0 80.0%

      \[\leadsto \color{blue}{y + 0.16666666666666666 \cdot {y}^{3}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 620:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+104}:\\ \;\;\;\;y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+155}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{3}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+180}:\\ \;\;\;\;y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;y + 0.16666666666666666 \cdot {y}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\\ \mathbf{if}\;y \leq 1400:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+155}:\\ \;\;\;\;\frac{0.16666666666666666 \cdot \left(x \cdot {y}^{3}\right)}{x}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+180}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y + 0.16666666666666666 \cdot {y}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (* -0.16666666666666666 (* y (pow x 2.0))))))
   (if (<= y 1400.0)
     (/ y (/ x (sin x)))
     (if (<= y 6e+46)
       t_0
       (if (<= y 1.35e+155)
         (/ (* 0.16666666666666666 (* x (pow y 3.0))) x)
         (if (<= y 4.1e+180)
           t_0
           (+ y (* 0.16666666666666666 (pow y 3.0)))))))))

C

double code(double x, double y) {
	double t_0 = y + (-0.16666666666666666 * (y * pow(x, 2.0)));
	double tmp;
	if (y <= 1400.0) {
		tmp = y / (x / sin(x));
	} else if (y <= 6e+46) {
		tmp = t_0;
	} else if (y <= 1.35e+155) {
		tmp = (0.16666666666666666 * (x * pow(y, 3.0))) / x;
	} else if (y <= 4.1e+180) {
		tmp = t_0;
	} else {
		tmp = y + (0.16666666666666666 * pow(y, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + ((-0.16666666666666666d0) * (y * (x ** 2.0d0)))
    if (y <= 1400.0d0) then
        tmp = y / (x / sin(x))
    else if (y <= 6d+46) then
        tmp = t_0
    else if (y <= 1.35d+155) then
        tmp = (0.16666666666666666d0 * (x * (y ** 3.0d0))) / x
    else if (y <= 4.1d+180) then
        tmp = t_0
    else
        tmp = y + (0.16666666666666666d0 * (y ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y + (-0.16666666666666666 * (y * Math.pow(x, 2.0)));
	double tmp;
	if (y <= 1400.0) {
		tmp = y / (x / Math.sin(x));
	} else if (y <= 6e+46) {
		tmp = t_0;
	} else if (y <= 1.35e+155) {
		tmp = (0.16666666666666666 * (x * Math.pow(y, 3.0))) / x;
	} else if (y <= 4.1e+180) {
		tmp = t_0;
	} else {
		tmp = y + (0.16666666666666666 * Math.pow(y, 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (-0.16666666666666666 * (y * math.pow(x, 2.0)))
	tmp = 0
	if y <= 1400.0:
		tmp = y / (x / math.sin(x))
	elif y <= 6e+46:
		tmp = t_0
	elif y <= 1.35e+155:
		tmp = (0.16666666666666666 * (x * math.pow(y, 3.0))) / x
	elif y <= 4.1e+180:
		tmp = t_0
	else:
		tmp = y + (0.16666666666666666 * math.pow(y, 3.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(-0.16666666666666666 * Float64(y * (x ^ 2.0))))
	tmp = 0.0
	if (y <= 1400.0)
		tmp = Float64(y / Float64(x / sin(x)));
	elseif (y <= 6e+46)
		tmp = t_0;
	elseif (y <= 1.35e+155)
		tmp = Float64(Float64(0.16666666666666666 * Float64(x * (y ^ 3.0))) / x);
	elseif (y <= 4.1e+180)
		tmp = t_0;
	else
		tmp = Float64(y + Float64(0.16666666666666666 * (y ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (-0.16666666666666666 * (y * (x ^ 2.0)));
	tmp = 0.0;
	if (y <= 1400.0)
		tmp = y / (x / sin(x));
	elseif (y <= 6e+46)
		tmp = t_0;
	elseif (y <= 1.35e+155)
		tmp = (0.16666666666666666 * (x * (y ^ 3.0))) / x;
	elseif (y <= 4.1e+180)
		tmp = t_0;
	else
		tmp = y + (0.16666666666666666 * (y ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(-0.16666666666666666 * N[(y * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1400.0], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+46], t$95$0, If[LessEqual[y, 1.35e+155], N[(N[(0.16666666666666666 * N[(x * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 4.1e+180], t$95$0, N[(y + N[(0.16666666666666666 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1400

   1. Initial program 83.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 49.8%

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   6. Step-by-step derivation

      1. associate-/l* 66.5%

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   7. Simplified 66.5%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]
   8. Step-by-step derivation

      1. clear-num 66.5%

         \[\leadsto y \cdot \color{blue}{\frac{1}{\frac{x}{\sin x}}} \]

      2. un-div-inv 66.5%

         \[\leadsto \color{blue}{\frac{y}{\frac{x}{\sin x}}} \]

   9. Applied egg-rr 66.5%

      \[\leadsto \color{blue}{\frac{y}{\frac{x}{\sin x}}} \]

   if 1400 < y < 6.00000000000000047e46 or 1.34999999999999997e155 < y < 4.1e180

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 3.0%

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   6. Step-by-step derivation

      1. associate-/l* 3.0%

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   7. Simplified 3.0%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   8. Taylor expanded in x around 0 51.4%

      \[\leadsto \color{blue}{y + -0.16666666666666666 \cdot \left({x}^{2} \cdot y\right)} \]

   if 6.00000000000000047e46 < y < 1.34999999999999997e155

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 83.4%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y + 0.16666666666666666 \cdot {y}^{3}\right)}}{x} \]

   4. Taylor expanded in x around 0 65.6%

      \[\leadsto \frac{\color{blue}{x \cdot \left(y + 0.16666666666666666 \cdot {y}^{3}\right)}}{x} \]

   5. Taylor expanded in y around inf 65.6%

      \[\leadsto \frac{\color{blue}{0.16666666666666666 \cdot \left(x \cdot {y}^{3}\right)}}{x} \]

   if 4.1e180 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 100.0%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y + 0.16666666666666666 \cdot {y}^{3}\right)}}{x} \]

   4. Taylor expanded in x around 0 80.0%

      \[\leadsto \color{blue}{y + 0.16666666666666666 \cdot {y}^{3}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1400:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+46}:\\ \;\;\;\;y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+155}:\\ \;\;\;\;\frac{0.16666666666666666 \cdot \left(x \cdot {y}^{3}\right)}{x}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+180}:\\ \;\;\;\;y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;y + 0.16666666666666666 \cdot {y}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+86}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y + 0.16666666666666666 \cdot {y}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e+86)
   (* (sin x) (/ y x))
   (+ y (* 0.16666666666666666 (pow y 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+86) {
		tmp = sin(x) * (y / x);
	} else {
		tmp = y + (0.16666666666666666 * pow(y, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.7d+86) then
        tmp = sin(x) * (y / x)
    else
        tmp = y + (0.16666666666666666d0 * (y ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+86) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = y + (0.16666666666666666 * Math.pow(y, 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.7e+86:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = y + (0.16666666666666666 * math.pow(y, 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.7e+86)
		tmp = Float64(sin(x) * Float64(y / x));
	else
		tmp = Float64(y + Float64(0.16666666666666666 * (y ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.7e+86)
		tmp = sin(x) * (y / x);
	else
		tmp = y + (0.16666666666666666 * (y ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.7e+86], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(y + N[(0.16666666666666666 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.70000000000000018e86

   1. Initial program 84.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 71.5%

      \[\leadsto \sin x \cdot \color{blue}{\frac{y}{x}} \]

   if 2.70000000000000018e86 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.2%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y + 0.16666666666666666 \cdot {y}^{3}\right)}}{x} \]

   4. Taylor expanded in x around 0 66.9%

      \[\leadsto \color{blue}{y + 0.16666666666666666 \cdot {y}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+86}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y + 0.16666666666666666 \cdot {y}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.1 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.1e+46) (* y (/ (sin x) x)) (* 0.16666666666666666 (pow y 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.1e+46) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = 0.16666666666666666 * pow(y, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.1d+46) then
        tmp = y * (sin(x) / x)
    else
        tmp = 0.16666666666666666d0 * (y ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.1e+46) {
		tmp = y * (Math.sin(x) / x);
	} else {
		tmp = 0.16666666666666666 * Math.pow(y, 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.1e+46:
		tmp = y * (math.sin(x) / x)
	else:
		tmp = 0.16666666666666666 * math.pow(y, 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.1e+46)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(0.16666666666666666 * (y ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.1e+46)
		tmp = y * (sin(x) / x);
	else
		tmp = 0.16666666666666666 * (y ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.1e+46], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0999999999999997e46

   1. Initial program 84.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 47.5%

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   6. Step-by-step derivation

      1. associate-/l* 63.4%

         \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   7. Simplified 63.4%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

   if 5.0999999999999997e46 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 94.7%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y + 0.16666666666666666 \cdot {y}^{3}\right)}}{x} \]

   4. Taylor expanded in x around 0 66.3%

      \[\leadsto \frac{\color{blue}{x \cdot \left(y + 0.16666666666666666 \cdot {y}^{3}\right)}}{x} \]

   5. Taylor expanded in y around inf 64.5%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot {y}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.1 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.85 \cdot 10^{+86}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.85e+86) (* (sin x) (/ y x)) (* 0.16666666666666666 (pow y 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.85e+86) {
		tmp = sin(x) * (y / x);
	} else {
		tmp = 0.16666666666666666 * pow(y, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.85d+86) then
        tmp = sin(x) * (y / x)
    else
        tmp = 0.16666666666666666d0 * (y ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.85e+86) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = 0.16666666666666666 * Math.pow(y, 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.85e+86:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = 0.16666666666666666 * math.pow(y, 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.85e+86)
		tmp = Float64(sin(x) * Float64(y / x));
	else
		tmp = Float64(0.16666666666666666 * (y ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.85e+86)
		tmp = sin(x) * (y / x);
	else
		tmp = 0.16666666666666666 * (y ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.85e+86], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.85e86

   1. Initial program 84.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 71.5%

      \[\leadsto \sin x \cdot \color{blue}{\frac{y}{x}} \]

   if 2.85e86 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.2%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y + 0.16666666666666666 \cdot {y}^{3}\right)}}{x} \]

   4. Taylor expanded in x around 0 66.9%

      \[\leadsto \frac{\color{blue}{x \cdot \left(y + 0.16666666666666666 \cdot {y}^{3}\right)}}{x} \]

   5. Taylor expanded in y around inf 66.9%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot {y}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.85 \cdot 10^{+86}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.85 \cdot 10^{+86}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.85e+86) (* x (/ y x)) (* 0.16666666666666666 (pow y 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.85e+86) {
		tmp = x * (y / x);
	} else {
		tmp = 0.16666666666666666 * pow(y, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.85d+86) then
        tmp = x * (y / x)
    else
        tmp = 0.16666666666666666d0 * (y ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.85e+86) {
		tmp = x * (y / x);
	} else {
		tmp = 0.16666666666666666 * Math.pow(y, 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.85e+86:
		tmp = x * (y / x)
	else:
		tmp = 0.16666666666666666 * math.pow(y, 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.85e+86)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(0.16666666666666666 * (y ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.85e+86)
		tmp = x * (y / x);
	else
		tmp = 0.16666666666666666 * (y ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.85e+86], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.85e86

   1. Initial program 84.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 47.1%

      \[\leadsto \frac{\sin x \cdot \color{blue}{y}}{x} \]

   4. Taylor expanded in x around 0 19.6%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{x} \]
   5. Step-by-step derivation

      1. *-commutative 19.6%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{x} \]

   6. Simplified 19.6%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{x} \]
   7. Step-by-step derivation

      1. *-commutative 19.6%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{x} \]

      2. associate-/l* 54.6%

         \[\leadsto \color{blue}{x \cdot \frac{y}{x}} \]

   8. Applied egg-rr 54.6%

      \[\leadsto \color{blue}{x \cdot \frac{y}{x}} \]

   if 2.85e86 < y

   1. Initial program 100.0%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.2%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y + 0.16666666666666666 \cdot {y}^{3}\right)}}{x} \]

   4. Taylor expanded in x around 0 66.9%

      \[\leadsto \frac{\color{blue}{x \cdot \left(y + 0.16666666666666666 \cdot {y}^{3}\right)}}{x} \]

   5. Taylor expanded in y around inf 66.9%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot {y}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.85 \cdot 10^{+86}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.3% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (/ y x)))

C

double code(double x, double y) {
	return x * (y / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (y / x)
end function

Java

public static double code(double x, double y) {
	return x * (y / x);
}

Python

def code(x, y):
	return x * (y / x)

Julia

function code(x, y)
	return Float64(x * Float64(y / x))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (y / x);
end

Wolfram

code[x_, y_] := N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.3%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Add Preprocessing

3. Taylor expanded in y around 0 38.7%

   \[\leadsto \frac{\sin x \cdot \color{blue}{y}}{x} \]

4. Taylor expanded in x around 0 19.1%

   \[\leadsto \frac{\color{blue}{x \cdot y}}{x} \]
5. Step-by-step derivation

   1. *-commutative 19.1%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{x} \]

6. Simplified 19.1%

   \[\leadsto \frac{\color{blue}{y \cdot x}}{x} \]
7. Step-by-step derivation

   1. *-commutative 19.1%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{x} \]

   2. associate-/l* 50.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{x}} \]

8. Applied egg-rr 50.8%

   \[\leadsto \color{blue}{x \cdot \frac{y}{x}} \]

9. Final simplification 50.8%

   \[\leadsto x \cdot \frac{y}{x} \]
10. Add Preprocessing

Alternative 10: 28.4% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 y)

C

double code(double x, double y) {
	return y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y
end function

Java

public static double code(double x, double y) {
	return y;
}

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

function tmp = code(x, y)
	tmp = y;
end

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 87.3%

   \[\frac{\sin x \cdot \sinh y}{x} \]
2. Step-by-step derivation

   1. associate-/l* 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 38.7%

   \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
6. Step-by-step derivation

   1. associate-/l* 51.3%

      \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

7. Simplified 51.3%

   \[\leadsto \color{blue}{y \cdot \frac{\sin x}{x}} \]

8. Taylor expanded in x around 0 26.3%

   \[\leadsto \color{blue}{y} \]

9. Final simplification 26.3%

   \[\leadsto y \]
10. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{\sinh y}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sin x) (/ (sinh y) x)))

C

double code(double x, double y) {
	return sin(x) * (sinh(y) / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(x) * (sinh(y) / x)
end function

Java

public static double code(double x, double y) {
	return Math.sin(x) * (Math.sinh(y) / x);
}

Python

def code(x, y):
	return math.sin(x) * (math.sinh(y) / x)

Julia

function code(x, y)
	return Float64(sin(x) * Float64(sinh(y) / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(x) * (sinh(y) / x);
end

Wolfram

code[x_, y_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024043 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (* (sin x) (/ (sinh y) x))

  (/ (* (sin x) (sinh y)) x))