Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.9% → 99.8%

Time: 6.6s

Alternatives: 9

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: 88.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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: 57.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4600:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+149}:\\ \;\;\;\;\frac{x \cdot \left(y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{1}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4600.0)
   (* y (/ (sin x) x))
   (if (<= y 1.9e+149)
     (/ (* x (+ y (* -0.16666666666666666 (* y (pow x 2.0))))) x)
     (* (sin x) (/ 1.0 (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4600.0) {
		tmp = y * (sin(x) / x);
	} else if (y <= 1.9e+149) {
		tmp = (x * (y + (-0.16666666666666666 * (y * pow(x, 2.0))))) / x;
	} else {
		tmp = sin(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 <= 4600.0d0) then
        tmp = y * (sin(x) / x)
    else if (y <= 1.9d+149) then
        tmp = (x * (y + ((-0.16666666666666666d0) * (y * (x ** 2.0d0))))) / x
    else
        tmp = sin(x) * (1.0d0 / (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4600.0) {
		tmp = y * (Math.sin(x) / x);
	} else if (y <= 1.9e+149) {
		tmp = (x * (y + (-0.16666666666666666 * (y * Math.pow(x, 2.0))))) / x;
	} else {
		tmp = Math.sin(x) * (1.0 / (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4600.0:
		tmp = y * (math.sin(x) / x)
	elif y <= 1.9e+149:
		tmp = (x * (y + (-0.16666666666666666 * (y * math.pow(x, 2.0))))) / x
	else:
		tmp = math.sin(x) * (1.0 / (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4600.0)
		tmp = Float64(y * Float64(sin(x) / x));
	elseif (y <= 1.9e+149)
		tmp = Float64(Float64(x * Float64(y + Float64(-0.16666666666666666 * Float64(y * (x ^ 2.0))))) / x);
	else
		tmp = Float64(sin(x) * Float64(1.0 / Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4600.0)
		tmp = y * (sin(x) / x);
	elseif (y <= 1.9e+149)
		tmp = (x * (y + (-0.16666666666666666 * (y * (x ^ 2.0))))) / x;
	else
		tmp = sin(x) * (1.0 / (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4600.0], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e+149], N[(N[(x * N[(y + N[(-0.16666666666666666 * N[(y * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4600

   1. Initial program 85.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 53.9%

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

      1. associate-/l* 68.1%

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

   7. Simplified 68.1%

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

   if 4600 < y < 1.9e149

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 3.1%

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

   4. Taylor expanded in x around 0 26.1%

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

   if 1.9e149 < 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 5.2%

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

      1. associate-/l* 5.2%

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

   7. Simplified 5.2%

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

      1. associate-*r/ 5.2%

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

      2. *-commutative 5.2%

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

      3. associate-*r/ 37.4%

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

      4. clear-num 37.4%

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

      5. un-div-inv 34.4%

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

   9. Applied egg-rr 34.4%

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

       1. frac-2neg 34.4%

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

       2. div-inv 37.4%

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

       3. distribute-neg-frac2 37.4%

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

   11. Applied egg-rr 37.4%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4600:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+149}:\\ \;\;\;\;\frac{x \cdot \left(y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{1}{\frac{x}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4600:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+148}:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot {x}^{2} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4600.0)
   (* y (/ (sin x) x))
   (if (<= y 3e+148)
     (* y (+ (* -0.16666666666666666 (pow x 2.0)) 1.0))
     (* (sin x) (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4600.0) {
		tmp = y * (sin(x) / x);
	} else if (y <= 3e+148) {
		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 <= 4600.0d0) then
        tmp = y * (sin(x) / x)
    else if (y <= 3d+148) 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 <= 4600.0) {
		tmp = y * (Math.sin(x) / x);
	} else if (y <= 3e+148) {
		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 <= 4600.0:
		tmp = y * (math.sin(x) / x)
	elif y <= 3e+148:
		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 <= 4600.0)
		tmp = Float64(y * Float64(sin(x) / x));
	elseif (y <= 3e+148)
		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 <= 4600.0)
		tmp = y * (sin(x) / x);
	elseif (y <= 3e+148)
		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, 4600.0], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e+148], 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 < 4600

   1. Initial program 85.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 53.9%

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

      1. associate-/l* 68.1%

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

   7. Simplified 68.1%

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

   if 4600 < y < 3.00000000000000015e148

   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 3.1%

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

      1. associate-/l* 3.1%

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

   7. Simplified 3.1%

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

   8. Taylor expanded in x around 0 20.5%

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

   if 3.00000000000000015e148 < 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 5.2%

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

      1. *-commutative 5.2%

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

      2. associate-/l* 37.4%

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

   7. Simplified 37.4%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4600:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+148}:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot {x}^{2} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4600:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+147}:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot {x}^{2} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{1}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4600.0)
   (* y (/ (sin x) x))
   (if (<= y 7e+147)
     (* y (+ (* -0.16666666666666666 (pow x 2.0)) 1.0))
     (* (sin x) (/ 1.0 (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4600.0) {
		tmp = y * (sin(x) / x);
	} else if (y <= 7e+147) {
		tmp = y * ((-0.16666666666666666 * pow(x, 2.0)) + 1.0);
	} else {
		tmp = sin(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 <= 4600.0d0) then
        tmp = y * (sin(x) / x)
    else if (y <= 7d+147) then
        tmp = y * (((-0.16666666666666666d0) * (x ** 2.0d0)) + 1.0d0)
    else
        tmp = sin(x) * (1.0d0 / (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4600.0) {
		tmp = y * (Math.sin(x) / x);
	} else if (y <= 7e+147) {
		tmp = y * ((-0.16666666666666666 * Math.pow(x, 2.0)) + 1.0);
	} else {
		tmp = Math.sin(x) * (1.0 / (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4600.0:
		tmp = y * (math.sin(x) / x)
	elif y <= 7e+147:
		tmp = y * ((-0.16666666666666666 * math.pow(x, 2.0)) + 1.0)
	else:
		tmp = math.sin(x) * (1.0 / (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4600.0)
		tmp = Float64(y * Float64(sin(x) / x));
	elseif (y <= 7e+147)
		tmp = Float64(y * Float64(Float64(-0.16666666666666666 * (x ^ 2.0)) + 1.0));
	else
		tmp = Float64(sin(x) * Float64(1.0 / Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4600.0)
		tmp = y * (sin(x) / x);
	elseif (y <= 7e+147)
		tmp = y * ((-0.16666666666666666 * (x ^ 2.0)) + 1.0);
	else
		tmp = sin(x) * (1.0 / (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4600.0], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7e+147], N[(y * N[(N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4600

   1. Initial program 85.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 53.9%

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

      1. associate-/l* 68.1%

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

   7. Simplified 68.1%

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

   if 4600 < y < 6.99999999999999949e147

   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 3.1%

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

      1. associate-/l* 3.1%

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

   7. Simplified 3.1%

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

   8. Taylor expanded in x around 0 20.5%

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

   if 6.99999999999999949e147 < 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 5.2%

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

      1. associate-/l* 5.2%

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

   7. Simplified 5.2%

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

      1. associate-*r/ 5.2%

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

      2. *-commutative 5.2%

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

      3. associate-*r/ 37.4%

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

      4. clear-num 37.4%

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

      5. un-div-inv 34.4%

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

   9. Applied egg-rr 34.4%

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

       1. frac-2neg 34.4%

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

       2. div-inv 37.4%

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

       3. distribute-neg-frac2 37.4%

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

   11. Applied egg-rr 37.4%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4600:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+147}:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot {x}^{2} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{1}{\frac{x}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 120000:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+149}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 120000.0)
   (* y (/ (sin x) x))
   (if (<= y 2.2e+149)
     (* -0.16666666666666666 (* y (pow x 2.0)))
     (* (sin x) (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 120000.0) {
		tmp = y * (sin(x) / x);
	} else if (y <= 2.2e+149) {
		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 <= 120000.0d0) then
        tmp = y * (sin(x) / x)
    else if (y <= 2.2d+149) 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 <= 120000.0) {
		tmp = y * (Math.sin(x) / x);
	} else if (y <= 2.2e+149) {
		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 <= 120000.0:
		tmp = y * (math.sin(x) / x)
	elif y <= 2.2e+149:
		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 <= 120000.0)
		tmp = Float64(y * Float64(sin(x) / x));
	elseif (y <= 2.2e+149)
		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 <= 120000.0)
		tmp = y * (sin(x) / x);
	elseif (y <= 2.2e+149)
		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, 120000.0], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+149], 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 < 1.2e5

   1. Initial program 85.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 53.9%

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

      1. associate-/l* 68.1%

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

   7. Simplified 68.1%

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

   if 1.2e5 < y < 2.2e149

   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 3.1%

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

      1. associate-/l* 3.1%

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

   7. Simplified 3.1%

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

   8. Taylor expanded in x around 0 20.5%

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

   9. Taylor expanded in x around inf 19.5%

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

   if 2.2e149 < 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 5.2%

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

      1. *-commutative 5.2%

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

      2. associate-/l* 37.4%

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

   7. Simplified 37.4%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 120000:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+149}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-8}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1e-8) (* x (/ y x)) (* y (/ (sin x) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1e-8) {
		tmp = x * (y / x);
	} else {
		tmp = y * (sin(x) / 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-8) then
        tmp = x * (y / x)
    else
        tmp = y * (sin(x) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1e-8) {
		tmp = x * (y / x);
	} else {
		tmp = y * (Math.sin(x) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1e-8:
		tmp = x * (y / x)
	else:
		tmp = y * (math.sin(x) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1e-8)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(y * Float64(sin(x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1e-8)
		tmp = x * (y / x);
	else
		tmp = y * (sin(x) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1e-8], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e-8

   1. Initial program 85.5%

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

   3. Taylor expanded in y around 0 36.3%

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

   4. Taylor expanded in x around 0 25.8%

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

      1. associate-/l* 54.5%

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

      2. *-commutative 54.5%

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

   6. Applied egg-rr 54.5%

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

   if 1e-8 < x

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 55.2%

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

      1. associate-/l* 55.3%

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

   7. Simplified 55.3%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-8}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.4% 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 89.3%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 41.2%

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

   1. *-commutative 41.2%

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

   2. associate-/l* 62.4%

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

7. Simplified 62.4%

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

8. Final simplification 62.4%

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

Alternative 8: 49.7% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in y around 0 41.2%

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

4. Taylor expanded in x around 0 22.4%

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

   1. associate-/l* 46.5%

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

   2. *-commutative 46.5%

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

6. Applied egg-rr 46.5%

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

7. Final simplification 46.5%

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

Alternative 9: 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 89.3%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 41.2%

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

   1. associate-/l* 51.9%

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

7. Simplified 51.9%

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

8. Taylor expanded in x around 0 27.5%

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

9. Final simplification 27.5%

   \[\leadsto y \]
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 2024095 
(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))