Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.1% → 99.8%

Time: 9.3s

Alternatives: 12

Speedup: 1.0×

• 50.4% 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: 89.1% 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 90.0%

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

   1. associate-/l* 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 2: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 360:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 360.0) (* y (/ (sin x) x)) (log1p (expm1 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 360.0) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = log1p(expm1(y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 360.0) {
		tmp = y * (Math.sin(x) / x);
	} else {
		tmp = Math.log1p(Math.expm1(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 360.0:
		tmp = y * (math.sin(x) / x)
	else:
		tmp = math.log1p(math.expm1(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 360.0)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = log1p(expm1(y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 360.0], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(Exp[y] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 360

   1. Initial program 86.3%

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

      1. associate-/l* 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 50.4%

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

      1. associate-/l* 64.0%

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

   7. Simplified 64.0%

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

   if 360 < y

   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 3.7%

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

      1. associate-/l* 3.7%

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

   7. Simplified 3.7%

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

      1. associate-*r/ 3.7%

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

      2. *-commutative 3.7%

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

      3. clear-num 3.7%

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

   9. Applied egg-rr 3.7%

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

   10. Taylor expanded in x around 0 3.5%

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

       1. remove-double-div 3.5%

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

       2. log1p-expm1-u 65.7%

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

   12. Applied egg-rr 65.7%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 360:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{+29}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+101}:\\ \;\;\;\;\frac{x \cdot \left(y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y + 0.16666666666666666 \cdot {y}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.6e+29)
   (* y (/ (sin x) x))
   (if (<= y 3.2e+101)
     (/ (* x (+ y (* -0.16666666666666666 (* y (pow x 2.0))))) x)
     (+ y (* 0.16666666666666666 (pow y 3.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.6e+29) {
		tmp = y * (sin(x) / x);
	} else if (y <= 3.2e+101) {
		tmp = (x * (y + (-0.16666666666666666 * (y * pow(x, 2.0))))) / 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 <= 4.6d+29) then
        tmp = y * (sin(x) / x)
    else if (y <= 3.2d+101) then
        tmp = (x * (y + ((-0.16666666666666666d0) * (y * (x ** 2.0d0))))) / 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 <= 4.6e+29) {
		tmp = y * (Math.sin(x) / x);
	} else if (y <= 3.2e+101) {
		tmp = (x * (y + (-0.16666666666666666 * (y * Math.pow(x, 2.0))))) / x;
	} else {
		tmp = y + (0.16666666666666666 * Math.pow(y, 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.6e+29:
		tmp = y * (math.sin(x) / x)
	elif y <= 3.2e+101:
		tmp = (x * (y + (-0.16666666666666666 * (y * math.pow(x, 2.0))))) / x
	else:
		tmp = y + (0.16666666666666666 * math.pow(y, 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.6e+29)
		tmp = Float64(y * Float64(sin(x) / x));
	elseif (y <= 3.2e+101)
		tmp = Float64(Float64(x * Float64(y + Float64(-0.16666666666666666 * Float64(y * (x ^ 2.0))))) / 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 <= 4.6e+29)
		tmp = y * (sin(x) / x);
	elseif (y <= 3.2e+101)
		tmp = (x * (y + (-0.16666666666666666 * (y * (x ^ 2.0))))) / x;
	else
		tmp = y + (0.16666666666666666 * (y ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.6e+29], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+101], N[(N[(x * N[(y + N[(-0.16666666666666666 * N[(y * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(y + N[(0.16666666666666666 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.6000000000000002e29

   1. Initial program 86.5%

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

      1. associate-/l* 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 49.7%

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

      1. associate-/l* 63.0%

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

   7. Simplified 63.0%

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

   if 4.6000000000000002e29 < y < 3.20000000000000005e101

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 2.7%

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

   4. Taylor expanded in x around 0 48.9%

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

   if 3.20000000000000005e101 < 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%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + 0.16666666666666666 \cdot {y}^{2}\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(1 \cdot y + \left(0.16666666666666666 \cdot {y}^{2}\right) \cdot y\right)}}{x} \]

      2. *-lft-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-plus 100.0%

         \[\leadsto \frac{\sin x \cdot \left(y + 0.16666666666666666 \cdot \color{blue}{{y}^{\left(2 + 1\right)}}\right)}{x} \]

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 71.4%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{+29}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+101}:\\ \;\;\;\;\frac{x \cdot \left(y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;y + 0.16666666666666666 \cdot {y}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+29}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+102}:\\ \;\;\;\;y \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;y + 0.16666666666666666 \cdot {y}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.4e+29)
   (* y (/ (sin x) x))
   (if (<= y 1.65e+102)
     (* y (+ 1.0 (* -0.16666666666666666 (pow x 2.0))))
     (+ y (* 0.16666666666666666 (pow y 3.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.4e+29) {
		tmp = y * (sin(x) / x);
	} else if (y <= 1.65e+102) {
		tmp = y * (1.0 + (-0.16666666666666666 * pow(x, 2.0)));
	} 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) :: tmp
    if (y <= 4.4d+29) then
        tmp = y * (sin(x) / x)
    else if (y <= 1.65d+102) then
        tmp = y * (1.0d0 + ((-0.16666666666666666d0) * (x ** 2.0d0)))
    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 <= 4.4e+29) {
		tmp = y * (Math.sin(x) / x);
	} else if (y <= 1.65e+102) {
		tmp = y * (1.0 + (-0.16666666666666666 * Math.pow(x, 2.0)));
	} else {
		tmp = y + (0.16666666666666666 * Math.pow(y, 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.4e+29:
		tmp = y * (math.sin(x) / x)
	elif y <= 1.65e+102:
		tmp = y * (1.0 + (-0.16666666666666666 * math.pow(x, 2.0)))
	else:
		tmp = y + (0.16666666666666666 * math.pow(y, 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.4e+29)
		tmp = Float64(y * Float64(sin(x) / x));
	elseif (y <= 1.65e+102)
		tmp = Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0))));
	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 <= 4.4e+29)
		tmp = y * (sin(x) / x);
	elseif (y <= 1.65e+102)
		tmp = y * (1.0 + (-0.16666666666666666 * (x ^ 2.0)));
	else
		tmp = y + (0.16666666666666666 * (y ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.4e+29], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.65e+102], N[(y * N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y + N[(0.16666666666666666 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.4000000000000003e29

   1. Initial program 86.5%

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

      1. associate-/l* 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 49.7%

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

      1. associate-/l* 63.0%

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

   7. Simplified 63.0%

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

   if 4.4000000000000003e29 < y < 1.64999999999999999e102

   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 2.7%

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

      1. associate-/l* 2.7%

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

   7. Simplified 2.7%

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

   8. Taylor expanded in x around 0 27.2%

      \[\leadsto y \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {x}^{2}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 27.2%

         \[\leadsto y \cdot \left(1 + \color{blue}{{x}^{2} \cdot -0.16666666666666666}\right) \]

   10. Simplified 27.2%

       \[\leadsto y \cdot \color{blue}{\left(1 + {x}^{2} \cdot -0.16666666666666666\right)} \]

   if 1.64999999999999999e102 < 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%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + 0.16666666666666666 \cdot {y}^{2}\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(1 \cdot y + \left(0.16666666666666666 \cdot {y}^{2}\right) \cdot y\right)}}{x} \]

      2. *-lft-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-plus 100.0%

         \[\leadsto \frac{\sin x \cdot \left(y + 0.16666666666666666 \cdot \color{blue}{{y}^{\left(2 + 1\right)}}\right)}{x} \]

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 71.4%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+29}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+102}:\\ \;\;\;\;y \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;y + 0.16666666666666666 \cdot {y}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-13}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 0.16666666666666666 \cdot {y}^{3}\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8.5e-13)
   (* y (/ (sin x) x))
   (/ (* x (+ y (* 0.16666666666666666 (pow y 3.0)))) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-13) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = (x * (y + (0.16666666666666666 * pow(y, 3.0)))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8.5d-13) then
        tmp = y * (sin(x) / x)
    else
        tmp = (x * (y + (0.16666666666666666d0 * (y ** 3.0d0)))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-13) {
		tmp = y * (Math.sin(x) / x);
	} else {
		tmp = (x * (y + (0.16666666666666666 * Math.pow(y, 3.0)))) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.5e-13:
		tmp = y * (math.sin(x) / x)
	else:
		tmp = (x * (y + (0.16666666666666666 * math.pow(y, 3.0)))) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.5e-13)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(Float64(x * Float64(y + Float64(0.16666666666666666 * (y ^ 3.0)))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.5e-13)
		tmp = y * (sin(x) / x);
	else
		tmp = (x * (y + (0.16666666666666666 * (y ^ 3.0)))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.5e-13], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y + N[(0.16666666666666666 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5000000000000001e-13

   1. Initial program 86.2%

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

      1. associate-/l* 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 50.6%

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

      1. associate-/l* 64.3%

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

   7. Simplified 64.3%

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

   if 8.5000000000000001e-13 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 60.8%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + 0.16666666666666666 \cdot {y}^{2}\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 60.8%

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(1 \cdot y + \left(0.16666666666666666 \cdot {y}^{2}\right) \cdot y\right)}}{x} \]

      2. *-lft-identity 60.8%

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

      3. associate-*l* 60.8%

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

      4. pow-plus 60.8%

         \[\leadsto \frac{\sin x \cdot \left(y + 0.16666666666666666 \cdot \color{blue}{{y}^{\left(2 + 1\right)}}\right)}{x} \]

      5. metadata-eval 60.8%

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

   5. Simplified 60.8%

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

   6. Taylor expanded in x around 0 48.8%

      \[\leadsto \frac{\color{blue}{x \cdot \left(y + 0.16666666666666666 \cdot {y}^{3}\right)}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-13}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + 0.16666666666666666 \cdot {y}^{3}\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+102}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y + 0.16666666666666666 \cdot {y}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.1e+102)
   (* (sin x) (/ y x))
   (+ y (* 0.16666666666666666 (pow y 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.1e+102) {
		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 <= 2.1d+102) 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 <= 2.1e+102) {
		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 <= 2.1e+102:
		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 <= 2.1e+102)
		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 <= 2.1e+102)
		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, 2.1e+102], 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 < 2.10000000000000001e102

   1. Initial program 88.1%

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

      1. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 44.2%

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

      1. *-commutative 44.2%

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

      2. associate-/l* 66.9%

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

   7. Simplified 66.9%

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

   if 2.10000000000000001e102 < 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%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + 0.16666666666666666 \cdot {y}^{2}\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(1 \cdot y + \left(0.16666666666666666 \cdot {y}^{2}\right) \cdot y\right)}}{x} \]

      2. *-lft-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-plus 100.0%

         \[\leadsto \frac{\sin x \cdot \left(y + 0.16666666666666666 \cdot \color{blue}{{y}^{\left(2 + 1\right)}}\right)}{x} \]

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 71.4%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+102}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y + 0.16666666666666666 \cdot {y}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 380:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 380.0) (* y (/ (sin x) x)) (* 0.16666666666666666 (pow y 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 380.0) {
		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 <= 380.0d0) 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 <= 380.0) {
		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 <= 380.0:
		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 <= 380.0)
		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 <= 380.0)
		tmp = y * (sin(x) / x);
	else
		tmp = 0.16666666666666666 * (y ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 380.0], 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 < 380

   1. Initial program 86.3%

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

      1. associate-/l* 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 50.4%

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

      1. associate-/l* 64.0%

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

   7. Simplified 64.0%

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

   if 380 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 61.5%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + 0.16666666666666666 \cdot {y}^{2}\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 61.5%

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(1 \cdot y + \left(0.16666666666666666 \cdot {y}^{2}\right) \cdot y\right)}}{x} \]

      2. *-lft-identity 61.5%

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

      3. associate-*l* 61.5%

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

      4. pow-plus 61.5%

         \[\leadsto \frac{\sin x \cdot \left(y + 0.16666666666666666 \cdot \color{blue}{{y}^{\left(2 + 1\right)}}\right)}{x} \]

      5. metadata-eval 61.5%

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

   5. Simplified 61.5%

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

   6. Taylor expanded in x around 0 44.0%

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

   7. Taylor expanded in y around inf 44.0%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 380:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.6e+102) (* (sin x) (/ y x)) (* 0.16666666666666666 (pow y 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.6e+102) {
		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 <= 5.6d+102) 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 <= 5.6e+102) {
		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 <= 5.6e+102:
		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 <= 5.6e+102)
		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 <= 5.6e+102)
		tmp = sin(x) * (y / x);
	else
		tmp = 0.16666666666666666 * (y ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.6e+102], 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 < 5.60000000000000037e102

   1. Initial program 88.1%

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

      1. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 44.2%

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

      1. *-commutative 44.2%

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

      2. associate-/l* 66.9%

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

   7. Simplified 66.9%

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

   if 5.60000000000000037e102 < 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%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + 0.16666666666666666 \cdot {y}^{2}\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(1 \cdot y + \left(0.16666666666666666 \cdot {y}^{2}\right) \cdot y\right)}}{x} \]

      2. *-lft-identity 100.0%

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

      3. associate-*l* 100.0%

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

      4. pow-plus 100.0%

         \[\leadsto \frac{\sin x \cdot \left(y + 0.16666666666666666 \cdot \color{blue}{{y}^{\left(2 + 1\right)}}\right)}{x} \]

      5. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 71.4%

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

   7. Taylor expanded in y around inf 71.4%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 100:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 100.0) y (* 0.16666666666666666 (pow y 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 100.0) {
		tmp = y;
	} 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 <= 100.0d0) then
        tmp = y
    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 <= 100.0) {
		tmp = y;
	} else {
		tmp = 0.16666666666666666 * Math.pow(y, 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 100.0:
		tmp = y
	else:
		tmp = 0.16666666666666666 * math.pow(y, 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 100.0)
		tmp = y;
	else
		tmp = Float64(0.16666666666666666 * (y ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 100.0)
		tmp = y;
	else
		tmp = 0.16666666666666666 * (y ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 100.0], y, N[(0.16666666666666666 * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 100

   1. Initial program 86.3%

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

      1. associate-/l* 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 50.4%

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

      1. associate-/l* 64.0%

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

   7. Simplified 64.0%

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

   8. Taylor expanded in x around 0 39.8%

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

   if 100 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 61.5%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\left(y \cdot \left(1 + 0.16666666666666666 \cdot {y}^{2}\right)\right)}}{x} \]
   4. Step-by-step derivation

      1. distribute-rgt-in 61.5%

         \[\leadsto \frac{\sin x \cdot \color{blue}{\left(1 \cdot y + \left(0.16666666666666666 \cdot {y}^{2}\right) \cdot y\right)}}{x} \]

      2. *-lft-identity 61.5%

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

      3. associate-*l* 61.5%

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

      4. pow-plus 61.5%

         \[\leadsto \frac{\sin x \cdot \left(y + 0.16666666666666666 \cdot \color{blue}{{y}^{\left(2 + 1\right)}}\right)}{x} \]

      5. metadata-eval 61.5%

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

   5. Simplified 61.5%

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

   6. Taylor expanded in x around 0 44.0%

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

   7. Taylor expanded in y around inf 44.0%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 100:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot {y}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 31.4% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+30}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{x \cdot y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5e+30) y (/ 1.0 (/ x (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e+30) {
		tmp = y;
	} else {
		tmp = 1.0 / (x / (x * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5d+30) then
        tmp = y
    else
        tmp = 1.0d0 / (x / (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5e+30) {
		tmp = y;
	} else {
		tmp = 1.0 / (x / (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5e+30:
		tmp = y
	else:
		tmp = 1.0 / (x / (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e+30)
		tmp = y;
	else
		tmp = Float64(1.0 / Float64(x / Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e+30)
		tmp = y;
	else
		tmp = 1.0 / (x / (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e+30], y, N[(1.0 / N[(x / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999998e30

   1. Initial program 86.5%

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

      1. associate-/l* 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 49.7%

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

      1. associate-/l* 63.0%

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

   7. Simplified 63.0%

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

   8. Taylor expanded in x around 0 39.2%

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

   if 4.9999999999999998e30 < y

   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 3.8%

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

      1. associate-/l* 3.8%

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

   7. Simplified 3.8%

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

      1. associate-*r/ 3.8%

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

      2. *-commutative 3.8%

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

      3. clear-num 3.8%

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

   9. Applied egg-rr 3.8%

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

   10. Taylor expanded in x around 0 18.9%

       \[\leadsto \frac{1}{\frac{x}{\color{blue}{x \cdot y}}} \]
   11. Step-by-step derivation

       1. *-commutative 18.9%

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

   12. Simplified 18.9%

       \[\leadsto \frac{1}{\frac{x}{\color{blue}{y \cdot x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+30}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{x \cdot y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 31.4% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+30}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3e+30) y (/ (* x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3e+30) {
		tmp = y;
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3d+30) then
        tmp = y
    else
        tmp = (x * y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3e+30) {
		tmp = y;
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3e+30:
		tmp = y
	else:
		tmp = (x * y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3e+30)
		tmp = y;
	else
		tmp = Float64(Float64(x * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3e+30)
		tmp = y;
	else
		tmp = (x * y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3e+30], y, N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.99999999999999978e30

   1. Initial program 86.5%

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

      1. associate-/l* 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 49.7%

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

      1. associate-/l* 63.0%

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

   7. Simplified 63.0%

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

   8. Taylor expanded in x around 0 39.2%

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

   if 2.99999999999999978e30 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 3.8%

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

   4. Taylor expanded in x around 0 18.9%

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

      1. *-commutative 18.9%

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

   6. Simplified 18.9%

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

4. Final simplification 33.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+30}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 28.0% 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 90.0%

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

   1. associate-/l* 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in y around 0 37.7%

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

   1. associate-/l* 47.5%

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

7. Simplified 47.5%

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

8. Taylor expanded in x around 0 29.9%

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

9. Final simplification 29.9%

   \[\leadsto y \]
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 2024071 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

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

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