Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 8.6s

Alternatives: 8

Speedup: 1.0×

• 49.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 81.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00048 \lor \neg \left(x \leq 6.5 \cdot 10^{+252}\right):\\ \;\;\;\;\sin y \cdot \left(\frac{1}{y} + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 0.00048) (not (<= x 6.5e+252)))
   (* (sin y) (+ (/ 1.0 y) (* 0.5 (* x (/ x y)))))
   (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 0.00048) || !(x <= 6.5e+252)) {
		tmp = sin(y) * ((1.0 / y) + (0.5 * (x * (x / y))));
	} else {
		tmp = cosh(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 <= 0.00048d0) .or. (.not. (x <= 6.5d+252))) then
        tmp = sin(y) * ((1.0d0 / y) + (0.5d0 * (x * (x / y))))
    else
        tmp = cosh(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= 0.00048) || !(x <= 6.5e+252)) {
		tmp = Math.sin(y) * ((1.0 / y) + (0.5 * (x * (x / y))));
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= 0.00048) or not (x <= 6.5e+252):
		tmp = math.sin(y) * ((1.0 / y) + (0.5 * (x * (x / y))))
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 0.00048) || !(x <= 6.5e+252))
		tmp = Float64(sin(y) * Float64(Float64(1.0 / y) + Float64(0.5 * Float64(x * Float64(x / y)))));
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= 0.00048) || ~((x <= 6.5e+252)))
		tmp = sin(y) * ((1.0 / y) + (0.5 * (x * (x / y))));
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 0.00048], N[Not[LessEqual[x, 6.5e+252]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] + N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.80000000000000012e-4 or 6.5e252 < x

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 89.6%

      \[\leadsto \sin y \cdot \color{blue}{\left(0.5 \cdot \frac{{x}^{2}}{y} + \frac{1}{y}\right)} \]
   6. Step-by-step derivation

      1. unpow2 89.6%

         \[\leadsto \sin y \cdot \left(0.5 \cdot \frac{\color{blue}{x \cdot x}}{y} + \frac{1}{y}\right) \]

      2. associate-/l* 85.8%

         \[\leadsto \sin y \cdot \left(0.5 \cdot \color{blue}{\left(x \cdot \frac{x}{y}\right)} + \frac{1}{y}\right) \]

   7. Applied egg-rr 85.8%

      \[\leadsto \sin y \cdot \left(0.5 \cdot \color{blue}{\left(x \cdot \frac{x}{y}\right)} + \frac{1}{y}\right) \]

   if 4.80000000000000012e-4 < x < 6.5e252

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 85.7%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

      1. *-rgt-identity 85.7%

         \[\leadsto \color{blue}{\cosh x} \]

      2. cosh-def 85.7%

         \[\leadsto \color{blue}{\frac{e^{x} + e^{-x}}{2}} \]

      3. clear-num 85.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{2}{e^{x} + e^{-x}}}} \]

      4. cosh-undef 85.7%

         \[\leadsto \frac{1}{\frac{2}{\color{blue}{2 \cdot \cosh x}}} \]

   5. Applied egg-rr 85.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{2 \cdot \cosh x}}} \]
   6. Step-by-step derivation

      1. associate-/r* 85.7%

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{2}{2}}{\cosh x}}} \]

      2. metadata-eval 85.7%

         \[\leadsto \frac{1}{\frac{\color{blue}{1}}{\cosh x}} \]

      3. remove-double-div 85.7%

         \[\leadsto \color{blue}{\cosh x} \]

   7. Simplified 85.7%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00048 \lor \neg \left(x \leq 6.5 \cdot 10^{+252}\right):\\ \;\;\;\;\sin y \cdot \left(\frac{1}{y} + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00048:\\ \;\;\;\;\sin y \cdot \left(\frac{1}{y} + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+127}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.00048)
   (* (sin y) (+ (/ 1.0 y) (* 0.5 (* x (/ x y)))))
   (if (<= x 3.8e+127)
     (cosh x)
     (* (sin y) (+ (/ 1.0 y) (* 0.5 (/ (* x x) y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.00048) {
		tmp = sin(y) * ((1.0 / y) + (0.5 * (x * (x / y))));
	} else if (x <= 3.8e+127) {
		tmp = cosh(x);
	} else {
		tmp = sin(y) * ((1.0 / y) + (0.5 * ((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 (x <= 0.00048d0) then
        tmp = sin(y) * ((1.0d0 / y) + (0.5d0 * (x * (x / y))))
    else if (x <= 3.8d+127) then
        tmp = cosh(x)
    else
        tmp = sin(y) * ((1.0d0 / y) + (0.5d0 * ((x * x) / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.00048) {
		tmp = Math.sin(y) * ((1.0 / y) + (0.5 * (x * (x / y))));
	} else if (x <= 3.8e+127) {
		tmp = Math.cosh(x);
	} else {
		tmp = Math.sin(y) * ((1.0 / y) + (0.5 * ((x * x) / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.00048:
		tmp = math.sin(y) * ((1.0 / y) + (0.5 * (x * (x / y))))
	elif x <= 3.8e+127:
		tmp = math.cosh(x)
	else:
		tmp = math.sin(y) * ((1.0 / y) + (0.5 * ((x * x) / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.00048)
		tmp = Float64(sin(y) * Float64(Float64(1.0 / y) + Float64(0.5 * Float64(x * Float64(x / y)))));
	elseif (x <= 3.8e+127)
		tmp = cosh(x);
	else
		tmp = Float64(sin(y) * Float64(Float64(1.0 / y) + Float64(0.5 * Float64(Float64(x * x) / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.00048)
		tmp = sin(y) * ((1.0 / y) + (0.5 * (x * (x / y))));
	elseif (x <= 3.8e+127)
		tmp = cosh(x);
	else
		tmp = sin(y) * ((1.0 / y) + (0.5 * ((x * x) / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.00048], N[(N[Sin[y], $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] + N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e+127], N[Cosh[x], $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] + N[(0.5 * N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.80000000000000012e-4

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 89.0%

      \[\leadsto \sin y \cdot \color{blue}{\left(0.5 \cdot \frac{{x}^{2}}{y} + \frac{1}{y}\right)} \]
   6. Step-by-step derivation

      1. unpow2 89.0%

         \[\leadsto \sin y \cdot \left(0.5 \cdot \frac{\color{blue}{x \cdot x}}{y} + \frac{1}{y}\right) \]

      2. associate-/l* 85.6%

         \[\leadsto \sin y \cdot \left(0.5 \cdot \color{blue}{\left(x \cdot \frac{x}{y}\right)} + \frac{1}{y}\right) \]

   7. Applied egg-rr 85.6%

      \[\leadsto \sin y \cdot \left(0.5 \cdot \color{blue}{\left(x \cdot \frac{x}{y}\right)} + \frac{1}{y}\right) \]

   if 4.80000000000000012e-4 < x < 3.7999999999999998e127

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 79.3%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

      1. *-rgt-identity 79.3%

         \[\leadsto \color{blue}{\cosh x} \]

      2. cosh-def 79.3%

         \[\leadsto \color{blue}{\frac{e^{x} + e^{-x}}{2}} \]

      3. clear-num 79.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{2}{e^{x} + e^{-x}}}} \]

      4. cosh-undef 79.3%

         \[\leadsto \frac{1}{\frac{2}{\color{blue}{2 \cdot \cosh x}}} \]

   5. Applied egg-rr 79.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{2 \cdot \cosh x}}} \]
   6. Step-by-step derivation

      1. associate-/r* 79.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{2}{2}}{\cosh x}}} \]

      2. metadata-eval 79.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{1}}{\cosh x}} \]

      3. remove-double-div 79.3%

         \[\leadsto \color{blue}{\cosh x} \]

   7. Simplified 79.3%

      \[\leadsto \color{blue}{\cosh x} \]

   if 3.7999999999999998e127 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 97.4%

      \[\leadsto \sin y \cdot \color{blue}{\left(0.5 \cdot \frac{{x}^{2}}{y} + \frac{1}{y}\right)} \]
   6. Step-by-step derivation

      1. unpow2 97.4%

         \[\leadsto \sin y \cdot \left(0.5 \cdot \frac{\color{blue}{x \cdot x}}{y} + \frac{1}{y}\right) \]

   7. Applied egg-rr 97.4%

      \[\leadsto \sin y \cdot \left(0.5 \cdot \frac{\color{blue}{x \cdot x}}{y} + \frac{1}{y}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00048:\\ \;\;\;\;\sin y \cdot \left(\frac{1}{y} + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+127}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00025:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 0.00025) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.00025) {
		tmp = sin(y) / y;
	} else {
		tmp = cosh(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 <= 0.00025d0) then
        tmp = sin(y) / y
    else
        tmp = cosh(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.00025) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.00025:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.00025)
		tmp = Float64(sin(y) / y);
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.00025)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.00025], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5000000000000001e-4

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 67.6%

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

   if 2.5000000000000001e-4 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 84.8%

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

      1. *-rgt-identity 84.8%

         \[\leadsto \color{blue}{\cosh x} \]

      2. cosh-def 84.8%

         \[\leadsto \color{blue}{\frac{e^{x} + e^{-x}}{2}} \]

      3. clear-num 84.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{2}{e^{x} + e^{-x}}}} \]

      4. cosh-undef 84.8%

         \[\leadsto \frac{1}{\frac{2}{\color{blue}{2 \cdot \cosh x}}} \]

   5. Applied egg-rr 84.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{2 \cdot \cosh x}}} \]
   6. Step-by-step derivation

      1. associate-/r* 84.8%

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{2}{2}}{\cosh x}}} \]

      2. metadata-eval 84.8%

         \[\leadsto \frac{1}{\frac{\color{blue}{1}}{\cosh x}} \]

      3. remove-double-div 84.8%

         \[\leadsto \color{blue}{\cosh x} \]

   7. Simplified 84.8%

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

Alternative 5: 63.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cosh x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (cosh x))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return math.cosh(x)

Julia

function code(x, y)
	return cosh(x)
end

MATLAB

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

Wolfram

code[x_, y_] := N[Cosh[x], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0 64.3%

   \[\leadsto \cosh x \cdot \color{blue}{1} \]
4. Step-by-step derivation

   1. *-rgt-identity 64.3%

      \[\leadsto \color{blue}{\cosh x} \]

   2. cosh-def 64.3%

      \[\leadsto \color{blue}{\frac{e^{x} + e^{-x}}{2}} \]

   3. clear-num 64.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{e^{x} + e^{-x}}}} \]

   4. cosh-undef 64.3%

      \[\leadsto \frac{1}{\frac{2}{\color{blue}{2 \cdot \cosh x}}} \]

5. Applied egg-rr 64.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{2}{2 \cdot \cosh x}}} \]
6. Step-by-step derivation

   1. associate-/r* 64.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{2}{2}}{\cosh x}}} \]

   2. metadata-eval 64.3%

      \[\leadsto \frac{1}{\frac{\color{blue}{1}}{\cosh x}} \]

   3. remove-double-div 64.3%

      \[\leadsto \color{blue}{\cosh x} \]

7. Simplified 64.3%

   \[\leadsto \color{blue}{\cosh x} \]
8. Add Preprocessing

Alternative 6: 52.0% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+249}:\\ \;\;\;\;y \cdot \left(\frac{1}{y} + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(y \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.3e+249)
   (* y (+ (/ 1.0 y) (* 0.5 (* x (/ x y)))))
   (* (/ 1.0 y) (* y (+ 1.0 (* -0.16666666666666666 (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.3e+249) {
		tmp = y * ((1.0 / y) + (0.5 * (x * (x / y))));
	} else {
		tmp = (1.0 / y) * (y * (1.0 + (-0.16666666666666666 * (y * 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 <= 3.3d+249) then
        tmp = y * ((1.0d0 / y) + (0.5d0 * (x * (x / y))))
    else
        tmp = (1.0d0 / y) * (y * (1.0d0 + ((-0.16666666666666666d0) * (y * y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.3e+249) {
		tmp = y * ((1.0 / y) + (0.5 * (x * (x / y))));
	} else {
		tmp = (1.0 / y) * (y * (1.0 + (-0.16666666666666666 * (y * y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.3e+249:
		tmp = y * ((1.0 / y) + (0.5 * (x * (x / y))))
	else:
		tmp = (1.0 / y) * (y * (1.0 + (-0.16666666666666666 * (y * y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.3e+249)
		tmp = Float64(y * Float64(Float64(1.0 / y) + Float64(0.5 * Float64(x * Float64(x / y)))));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.3e+249)
		tmp = y * ((1.0 / y) + (0.5 * (x * (x / y))));
	else
		tmp = (1.0 / y) * (y * (1.0 + (-0.16666666666666666 * (y * y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.3e+249], N[(y * N[(N[(1.0 / y), $MachinePrecision] + N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(y * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.30000000000000014e249

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.9%

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

      3. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 83.6%

      \[\leadsto \sin y \cdot \color{blue}{\left(0.5 \cdot \frac{{x}^{2}}{y} + \frac{1}{y}\right)} \]
   6. Step-by-step derivation

      1. unpow2 83.6%

         \[\leadsto \sin y \cdot \left(0.5 \cdot \frac{\color{blue}{x \cdot x}}{y} + \frac{1}{y}\right) \]

   7. Applied egg-rr 83.6%

      \[\leadsto \sin y \cdot \left(0.5 \cdot \frac{\color{blue}{x \cdot x}}{y} + \frac{1}{y}\right) \]
   8. Step-by-step derivation

      1. associate-/l* 79.4%

         \[\leadsto \sin y \cdot \left(0.5 \cdot \color{blue}{\left(x \cdot \frac{x}{y}\right)} + \frac{1}{y}\right) \]

      2. *-commutative 79.4%

         \[\leadsto \sin y \cdot \left(0.5 \cdot \color{blue}{\left(\frac{x}{y} \cdot x\right)} + \frac{1}{y}\right) \]

   9. Applied egg-rr 79.4%

      \[\leadsto \sin y \cdot \left(0.5 \cdot \color{blue}{\left(\frac{x}{y} \cdot x\right)} + \frac{1}{y}\right) \]

   10. Taylor expanded in y around 0 54.4%

       \[\leadsto \color{blue}{y} \cdot \left(0.5 \cdot \left(\frac{x}{y} \cdot x\right) + \frac{1}{y}\right) \]

   if 3.30000000000000014e249 < y

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l/ 99.6%

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

      3. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 50.6%

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

   6. Taylor expanded in y around 0 34.1%

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

      1. unpow2 34.1%

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

   8. Applied egg-rr 34.1%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+249}:\\ \;\;\;\;y \cdot \left(\frac{1}{y} + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \left(y \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.4% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(\frac{1}{y} + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* y (+ (/ 1.0 y) (* 0.5 (* x (/ x y))))))

C

double code(double x, double y) {
	return y * ((1.0 / y) + (0.5 * (x * (x / y))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y * ((1.0d0 / y) + (0.5d0 * (x * (x / y))))
end function

Java

public static double code(double x, double y) {
	return y * ((1.0 / y) + (0.5 * (x * (x / y))));
}

Python

def code(x, y):
	return y * ((1.0 / y) + (0.5 * (x * (x / y))))

Julia

function code(x, y)
	return Float64(y * Float64(Float64(1.0 / y) + Float64(0.5 * Float64(x * Float64(x / y)))))
end

MATLAB

function tmp = code(x, y)
	tmp = y * ((1.0 / y) + (0.5 * (x * (x / y))));
end

Wolfram

code[x_, y_] := N[(y * N[(N[(1.0 / y), $MachinePrecision] + N[(0.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-/l* 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 83.6%

   \[\leadsto \sin y \cdot \color{blue}{\left(0.5 \cdot \frac{{x}^{2}}{y} + \frac{1}{y}\right)} \]
6. Step-by-step derivation

   1. unpow2 83.6%

      \[\leadsto \sin y \cdot \left(0.5 \cdot \frac{\color{blue}{x \cdot x}}{y} + \frac{1}{y}\right) \]

7. Applied egg-rr 83.6%

   \[\leadsto \sin y \cdot \left(0.5 \cdot \frac{\color{blue}{x \cdot x}}{y} + \frac{1}{y}\right) \]
8. Step-by-step derivation

   1. associate-/l* 78.5%

      \[\leadsto \sin y \cdot \left(0.5 \cdot \color{blue}{\left(x \cdot \frac{x}{y}\right)} + \frac{1}{y}\right) \]

   2. *-commutative 78.5%

      \[\leadsto \sin y \cdot \left(0.5 \cdot \color{blue}{\left(\frac{x}{y} \cdot x\right)} + \frac{1}{y}\right) \]

9. Applied egg-rr 78.5%

   \[\leadsto \sin y \cdot \left(0.5 \cdot \color{blue}{\left(\frac{x}{y} \cdot x\right)} + \frac{1}{y}\right) \]

10. Taylor expanded in y around 0 52.3%

    \[\leadsto \color{blue}{y} \cdot \left(0.5 \cdot \left(\frac{x}{y} \cdot x\right) + \frac{1}{y}\right) \]

11. Final simplification 52.3%

    \[\leadsto y \cdot \left(\frac{1}{y} + 0.5 \cdot \left(x \cdot \frac{x}{y}\right)\right) \]
12. Add Preprocessing

Alternative 8: 26.8% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-*l/ 99.8%

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

   3. associate-/l* 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 50.8%

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

6. Taylor expanded in y around 0 23.7%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024132 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (* (cosh x) (sin y)) y))

  (* (cosh x) (/ (sin y) y)))