Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.1% → 99.9%

Time: 8.6s

Alternatives: 10

Speedup: 1.0×

• 51.0% 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: 99.9% 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 91.1%

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

Alternative 2: 67.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2500000000000:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+276} \lor \neg \left(y \leq 1.26 \cdot 10^{+297}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot {x}^{2} + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2500000000000.0)
   (* y (/ (sin x) x))
   (if (or (<= y 1.6e+276) (not (<= y 1.26e+297)))
     (log1p (expm1 y))
     (* y (+ (* -0.16666666666666666 (pow x 2.0)) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2500000000000.0) {
		tmp = y * (sin(x) / x);
	} else if ((y <= 1.6e+276) || !(y <= 1.26e+297)) {
		tmp = log1p(expm1(y));
	} else {
		tmp = y * ((-0.16666666666666666 * pow(x, 2.0)) + 1.0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2500000000000.0) {
		tmp = y * (Math.sin(x) / x);
	} else if ((y <= 1.6e+276) || !(y <= 1.26e+297)) {
		tmp = Math.log1p(Math.expm1(y));
	} else {
		tmp = y * ((-0.16666666666666666 * Math.pow(x, 2.0)) + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2500000000000.0:
		tmp = y * (math.sin(x) / x)
	elif (y <= 1.6e+276) or not (y <= 1.26e+297):
		tmp = math.log1p(math.expm1(y))
	else:
		tmp = y * ((-0.16666666666666666 * math.pow(x, 2.0)) + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2500000000000.0)
		tmp = Float64(y * Float64(sin(x) / x));
	elseif ((y <= 1.6e+276) || !(y <= 1.26e+297))
		tmp = log1p(expm1(y));
	else
		tmp = Float64(y * Float64(Float64(-0.16666666666666666 * (x ^ 2.0)) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2500000000000.0], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 1.6e+276], N[Not[LessEqual[y, 1.26e+297]], $MachinePrecision]], N[Log[1 + N[(Exp[y] - 1), $MachinePrecision]], $MachinePrecision], N[(y * N[(N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.5e12

   1. Initial program 88.2%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 51.0%

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

      1. associate-/l* 62.7%

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

   7. Simplified 62.7%

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

   if 2.5e12 < y < 1.59999999999999994e276 or 1.26e297 < 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 4.0%

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

      1. associate-/l* 4.0%

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

   7. Simplified 4.0%

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

      1. associate-*r/ 4.0%

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

      2. clear-num 4.0%

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

      3. *-commutative 4.0%

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

   9. Applied egg-rr 4.0%

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

   10. Taylor expanded in x around 0 3.8%

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

       1. remove-double-div 3.8%

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

       2. log1p-expm1-u 75.0%

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

   12. Applied egg-rr 75.0%

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

   if 1.59999999999999994e276 < y < 1.26e297

   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 6.1%

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

      1. associate-/l* 6.1%

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

   7. Simplified 6.1%

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

   8. Taylor expanded in x around 0 74.2%

      \[\leadsto y \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {x}^{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2500000000000:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+276} \lor \neg \left(y \leq 1.26 \cdot 10^{+297}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot {x}^{2} + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 13.2:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+63}:\\ \;\;\;\;\frac{x \cdot \left(y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\right)}{x}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+272}:\\ \;\;\;\;\sin x \cdot \frac{1}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+297}:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot {x}^{2} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 13.2)
   (* y (/ (sin x) x))
   (if (<= y 4e+63)
     (/ (* x (+ y (* -0.16666666666666666 (* y (pow x 2.0))))) x)
     (if (<= y 8.8e+272)
       (* (sin x) (/ 1.0 (/ x y)))
       (if (<= y 2e+297)
         (* y (+ (* -0.16666666666666666 (pow x 2.0)) 1.0))
         (/ (* x y) x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 13.2) {
		tmp = y * (sin(x) / x);
	} else if (y <= 4e+63) {
		tmp = (x * (y + (-0.16666666666666666 * (y * pow(x, 2.0))))) / x;
	} else if (y <= 8.8e+272) {
		tmp = sin(x) * (1.0 / (x / y));
	} else if (y <= 2e+297) {
		tmp = y * ((-0.16666666666666666 * pow(x, 2.0)) + 1.0);
	} 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 <= 13.2d0) then
        tmp = y * (sin(x) / x)
    else if (y <= 4d+63) then
        tmp = (x * (y + ((-0.16666666666666666d0) * (y * (x ** 2.0d0))))) / x
    else if (y <= 8.8d+272) then
        tmp = sin(x) * (1.0d0 / (x / y))
    else if (y <= 2d+297) then
        tmp = y * (((-0.16666666666666666d0) * (x ** 2.0d0)) + 1.0d0)
    else
        tmp = (x * y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 13.2) {
		tmp = y * (Math.sin(x) / x);
	} else if (y <= 4e+63) {
		tmp = (x * (y + (-0.16666666666666666 * (y * Math.pow(x, 2.0))))) / x;
	} else if (y <= 8.8e+272) {
		tmp = Math.sin(x) * (1.0 / (x / y));
	} else if (y <= 2e+297) {
		tmp = y * ((-0.16666666666666666 * Math.pow(x, 2.0)) + 1.0);
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 13.2:
		tmp = y * (math.sin(x) / x)
	elif y <= 4e+63:
		tmp = (x * (y + (-0.16666666666666666 * (y * math.pow(x, 2.0))))) / x
	elif y <= 8.8e+272:
		tmp = math.sin(x) * (1.0 / (x / y))
	elif y <= 2e+297:
		tmp = y * ((-0.16666666666666666 * math.pow(x, 2.0)) + 1.0)
	else:
		tmp = (x * y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 13.2)
		tmp = Float64(y * Float64(sin(x) / x));
	elseif (y <= 4e+63)
		tmp = Float64(Float64(x * Float64(y + Float64(-0.16666666666666666 * Float64(y * (x ^ 2.0))))) / x);
	elseif (y <= 8.8e+272)
		tmp = Float64(sin(x) * Float64(1.0 / Float64(x / y)));
	elseif (y <= 2e+297)
		tmp = Float64(y * Float64(Float64(-0.16666666666666666 * (x ^ 2.0)) + 1.0));
	else
		tmp = Float64(Float64(x * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 13.2)
		tmp = y * (sin(x) / x);
	elseif (y <= 4e+63)
		tmp = (x * (y + (-0.16666666666666666 * (y * (x ^ 2.0))))) / x;
	elseif (y <= 8.8e+272)
		tmp = sin(x) * (1.0 / (x / y));
	elseif (y <= 2e+297)
		tmp = y * ((-0.16666666666666666 * (x ^ 2.0)) + 1.0);
	else
		tmp = (x * y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 13.2], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+63], N[(N[(x * N[(y + N[(-0.16666666666666666 * N[(y * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 8.8e+272], N[(N[Sin[x], $MachinePrecision] * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+297], N[(y * N[(N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 13.199999999999999

   1. Initial program 88.0%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 51.4%

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

      1. associate-/l* 63.3%

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

   7. Simplified 63.3%

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

   if 13.199999999999999 < y < 4.00000000000000023e63

   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 3.5%

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

   6. Taylor expanded in x around 0 30.1%

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

   if 4.00000000000000023e63 < y < 8.80000000000000034e272

   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 4.3%

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

      1. associate-/l* 4.3%

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

   7. Simplified 4.3%

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

      1. associate-*r/ 4.3%

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

      2. *-commutative 4.3%

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

      3. associate-*r/ 28.9%

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

      4. clear-num 28.9%

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

      5. un-div-inv 28.9%

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

   9. Applied egg-rr 28.9%

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

       1. frac-2neg 28.9%

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

       2. div-inv 28.9%

          \[\leadsto \color{blue}{\left(-\sin x\right) \cdot \frac{1}{-\frac{x}{y}}} \]

       3. distribute-neg-frac2 28.9%

          \[\leadsto \left(-\sin x\right) \cdot \frac{1}{\color{blue}{\frac{x}{-y}}} \]

   11. Applied egg-rr 28.9%

       \[\leadsto \color{blue}{\left(-\sin x\right) \cdot \frac{1}{\frac{x}{-y}}} \]

   if 8.80000000000000034e272 < y < 2e297

   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 6.1%

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

      1. associate-/l* 6.1%

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

   7. Simplified 6.1%

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

   8. Taylor expanded in x around 0 74.2%

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

   if 2e297 < 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. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{x \cdot y}}{x} \]
3. Recombined 5 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 13.2:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+63}:\\ \;\;\;\;\frac{x \cdot \left(y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\right)}{x}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+272}:\\ \;\;\;\;\sin x \cdot \frac{1}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+297}:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot {x}^{2} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 13.2:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+63}:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot {x}^{2} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 13.2)
   (* y (/ (sin x) x))
   (if (<= y 3.7e+63)
     (* y (+ (* -0.16666666666666666 (pow x 2.0)) 1.0))
     (* (sin x) (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 13.2) {
		tmp = y * (sin(x) / x);
	} else if (y <= 3.7e+63) {
		tmp = y * ((-0.16666666666666666 * pow(x, 2.0)) + 1.0);
	} else {
		tmp = sin(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 <= 13.2d0) then
        tmp = y * (sin(x) / x)
    else if (y <= 3.7d+63) then
        tmp = y * (((-0.16666666666666666d0) * (x ** 2.0d0)) + 1.0d0)
    else
        tmp = sin(x) * (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 13.2) {
		tmp = y * (Math.sin(x) / x);
	} else if (y <= 3.7e+63) {
		tmp = y * ((-0.16666666666666666 * Math.pow(x, 2.0)) + 1.0);
	} else {
		tmp = Math.sin(x) * (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 13.2:
		tmp = y * (math.sin(x) / x)
	elif y <= 3.7e+63:
		tmp = y * ((-0.16666666666666666 * math.pow(x, 2.0)) + 1.0)
	else:
		tmp = math.sin(x) * (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 13.2)
		tmp = Float64(y * Float64(sin(x) / x));
	elseif (y <= 3.7e+63)
		tmp = Float64(y * Float64(Float64(-0.16666666666666666 * (x ^ 2.0)) + 1.0));
	else
		tmp = Float64(sin(x) * Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 13.2)
		tmp = y * (sin(x) / x);
	elseif (y <= 3.7e+63)
		tmp = y * ((-0.16666666666666666 * (x ^ 2.0)) + 1.0);
	else
		tmp = sin(x) * (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 13.2], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7e+63], N[(y * N[(N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 13.199999999999999

   1. Initial program 88.0%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 51.4%

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

      1. associate-/l* 63.3%

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

   7. Simplified 63.3%

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

   if 13.199999999999999 < y < 3.69999999999999968e63

   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 3.5%

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

      1. associate-/l* 3.5%

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

   7. Simplified 3.5%

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

   8. Taylor expanded in x around 0 30.1%

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

   if 3.69999999999999968e63 < 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 28.8%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 13.2:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+63}:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot {x}^{2} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1200:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{+63}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1200.0)
   (* y (/ (sin x) x))
   (if (<= y 1.16e+63)
     (* -0.16666666666666666 (* y (pow x 2.0)))
     (* (sin x) (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1200.0) {
		tmp = y * (sin(x) / x);
	} else if (y <= 1.16e+63) {
		tmp = -0.16666666666666666 * (y * pow(x, 2.0));
	} else {
		tmp = sin(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 <= 1200.0d0) then
        tmp = y * (sin(x) / x)
    else if (y <= 1.16d+63) then
        tmp = (-0.16666666666666666d0) * (y * (x ** 2.0d0))
    else
        tmp = sin(x) * (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1200.0) {
		tmp = y * (Math.sin(x) / x);
	} else if (y <= 1.16e+63) {
		tmp = -0.16666666666666666 * (y * Math.pow(x, 2.0));
	} else {
		tmp = Math.sin(x) * (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1200.0:
		tmp = y * (math.sin(x) / x)
	elif y <= 1.16e+63:
		tmp = -0.16666666666666666 * (y * math.pow(x, 2.0))
	else:
		tmp = math.sin(x) * (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1200.0)
		tmp = Float64(y * Float64(sin(x) / x));
	elseif (y <= 1.16e+63)
		tmp = Float64(-0.16666666666666666 * Float64(y * (x ^ 2.0)));
	else
		tmp = Float64(sin(x) * Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1200.0)
		tmp = y * (sin(x) / x);
	elseif (y <= 1.16e+63)
		tmp = -0.16666666666666666 * (y * (x ^ 2.0));
	else
		tmp = sin(x) * (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1200.0], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.16e+63], N[(-0.16666666666666666 * N[(y * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1200

   1. Initial program 88.1%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 51.2%

      \[\leadsto \color{blue}{\frac{y \cdot \sin x}{x}} \]
   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 1200 < y < 1.15999999999999994e63

   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 2.5%

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

      1. associate-/l* 2.5%

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

   7. Simplified 2.5%

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

   8. Taylor expanded in x around 0 31.9%

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

   9. Taylor expanded in x around inf 31.3%

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

   if 1.15999999999999994e63 < 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 28.8%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1200:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{+63}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+37}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \frac{-1}{x \cdot y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.3e+37) (* y (/ (sin x) x)) (/ -1.0 (* x (/ -1.0 (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.3e+37) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = -1.0 / (x * (-1.0 / (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 <= 1.3d+37) then
        tmp = y * (sin(x) / x)
    else
        tmp = (-1.0d0) / (x * ((-1.0d0) / (x * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.3e+37) {
		tmp = y * (Math.sin(x) / x);
	} else {
		tmp = -1.0 / (x * (-1.0 / (x * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.3e+37:
		tmp = y * (math.sin(x) / x)
	else:
		tmp = -1.0 / (x * (-1.0 / (x * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.3e+37)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = Float64(-1.0 / Float64(x * Float64(-1.0 / Float64(x * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.3e+37)
		tmp = y * (sin(x) / x);
	else
		tmp = -1.0 / (x * (-1.0 / (x * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.3e+37], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(x * N[(-1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3e37

   1. Initial program 88.4%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 50.0%

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

      1. associate-/l* 61.5%

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

   7. Simplified 61.5%

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

   if 1.3e37 < 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 4.4%

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

      1. associate-/l* 4.4%

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

   7. Simplified 4.4%

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

      1. associate-*r/ 4.4%

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

      2. clear-num 4.4%

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

      3. *-commutative 4.4%

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

   9. Applied egg-rr 4.4%

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

       1. clear-num 4.4%

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

       2. associate-/r/ 4.4%

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

       3. *-commutative 4.4%

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

       4. associate-/r* 4.4%

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

   11. Applied egg-rr 4.4%

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

   12. Taylor expanded in x around 0 16.6%

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

       1. *-commutative 16.6%

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

   14. Simplified 16.6%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+37}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \frac{-1}{x \cdot y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 61.0%

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

Alternative 8: 30.8% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+32}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \frac{-1}{x \cdot y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1e+32) y (/ -1.0 (* x (/ -1.0 (* x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1e+32) {
		tmp = y;
	} else {
		tmp = -1.0 / (x * (-1.0 / (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 (x <= 1d+32) then
        tmp = y
    else
        tmp = (-1.0d0) / (x * ((-1.0d0) / (x * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1e+32) {
		tmp = y;
	} else {
		tmp = -1.0 / (x * (-1.0 / (x * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1e+32:
		tmp = y
	else:
		tmp = -1.0 / (x * (-1.0 / (x * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1e+32)
		tmp = y;
	else
		tmp = Float64(-1.0 / Float64(x * Float64(-1.0 / Float64(x * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1e+32)
		tmp = y;
	else
		tmp = -1.0 / (x * (-1.0 / (x * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1e+32], y, N[(-1.0 / N[(x * N[(-1.0 / N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000005e32

   1. Initial program 88.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 36.2%

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

      1. associate-/l* 47.6%

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

   7. Simplified 47.6%

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

   8. Taylor expanded in x around 0 32.4%

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

   if 1.00000000000000005e32 < x

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 50.8%

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

      1. associate-/l* 50.8%

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

   7. Simplified 50.8%

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

      1. associate-*r/ 50.8%

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

      2. clear-num 50.4%

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

      3. *-commutative 50.4%

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

   9. Applied egg-rr 50.4%

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

       1. clear-num 50.4%

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

       2. associate-/r/ 50.4%

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

       3. *-commutative 50.4%

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

       4. associate-/r* 50.4%

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

   11. Applied egg-rr 50.4%

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

   12. Taylor expanded in x around 0 17.1%

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

       1. *-commutative 17.1%

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

   14. Simplified 17.1%

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

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+32}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \frac{-1}{x \cdot y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.8% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+33}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1e+33) y (/ (* x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1e+33) {
		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 (x <= 1d+33) 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 (x <= 1e+33) {
		tmp = y;
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1e+33:
		tmp = y
	else:
		tmp = (x * y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1e+33)
		tmp = y;
	else
		tmp = Float64(Float64(x * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1e+33)
		tmp = y;
	else
		tmp = (x * y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1e+33], y, N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999995e32

   1. Initial program 88.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 36.2%

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

      1. associate-/l* 47.6%

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

   7. Simplified 47.6%

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

   8. Taylor expanded in x around 0 32.4%

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

   if 9.9999999999999995e32 < x

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 50.8%

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

   6. Taylor expanded in x around 0 17.1%

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

Alternative 10: 27.5% 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 91.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 39.5%

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

   1. associate-/l* 48.3%

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

7. Simplified 48.3%

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

8. Taylor expanded in x around 0 26.1%

   \[\leadsto \color{blue}{y} \]
9. Add Preprocessing

Developer target: 99.9% 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 2024085 
(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))