Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 10.5s

Alternatives: 11

Speedup: 1.0×

• 48.8% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]

2. Final simplification 99.9%

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

Alternative 2: 71.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sin y \cdot \left(x \cdot x\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.5e-9)
   (/ (sin y) y)
   (if (<= x 1.4e+154) (cosh x) (* 0.5 (/ (* (sin y) (* x x)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.5e-9) {
		tmp = sin(y) / y;
	} else if (x <= 1.4e+154) {
		tmp = cosh(x);
	} else {
		tmp = 0.5 * ((sin(y) * (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 <= 3.5d-9) then
        tmp = sin(y) / y
    else if (x <= 1.4d+154) then
        tmp = cosh(x)
    else
        tmp = 0.5d0 * ((sin(y) * (x * x)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 3.5e-9) {
		tmp = Math.sin(y) / y;
	} else if (x <= 1.4e+154) {
		tmp = Math.cosh(x);
	} else {
		tmp = 0.5 * ((Math.sin(y) * (x * x)) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.5e-9:
		tmp = math.sin(y) / y
	elif x <= 1.4e+154:
		tmp = math.cosh(x)
	else:
		tmp = 0.5 * ((math.sin(y) * (x * x)) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.5e-9)
		tmp = Float64(sin(y) / y);
	elseif (x <= 1.4e+154)
		tmp = cosh(x);
	else
		tmp = Float64(0.5 * Float64(Float64(sin(y) * Float64(x * x)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.5e-9)
		tmp = sin(y) / y;
	elseif (x <= 1.4e+154)
		tmp = cosh(x);
	else
		tmp = 0.5 * ((sin(y) * (x * x)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.5e-9], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 1.4e+154], N[Cosh[x], $MachinePrecision], N[(0.5 * N[(N[(N[Sin[y], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.4999999999999999e-9

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 85.0%

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

      1. unpow2 85.0%

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

   4. Simplified 85.0%

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

      1. clear-num 85.0%

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

      2. associate-/r/ 84.9%

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

   6. Applied egg-rr 84.9%

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

      1. /-rgt-identity 84.9%

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

      2. associate-/r/ 85.0%

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

   8. Applied egg-rr 85.0%

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

   9. Taylor expanded in x around 0 69.3%

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

   if 3.4999999999999999e-9 < x < 1.4e154

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 75.1%

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

   if 1.4e154 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2} \cdot \sin y}{y}} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sin y \cdot \left(x \cdot x\right)}{y}\\ \end{array} \]

Alternative 3: 72.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+153}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \cosh x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sin y \cdot \left(x \cdot x\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.5e-9)
   (/ (sin y) y)
   (if (<= x 4e+153)
     (* (+ 1.0 (* -0.16666666666666666 (* y y))) (cosh x))
     (* 0.5 (/ (* (sin y) (* x x)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.5e-9) {
		tmp = sin(y) / y;
	} else if (x <= 4e+153) {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * cosh(x);
	} else {
		tmp = 0.5 * ((sin(y) * (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 <= 3.5d-9) then
        tmp = sin(y) / y
    else if (x <= 4d+153) then
        tmp = (1.0d0 + ((-0.16666666666666666d0) * (y * y))) * cosh(x)
    else
        tmp = 0.5d0 * ((sin(y) * (x * x)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 3.5e-9) {
		tmp = Math.sin(y) / y;
	} else if (x <= 4e+153) {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * Math.cosh(x);
	} else {
		tmp = 0.5 * ((Math.sin(y) * (x * x)) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.5e-9:
		tmp = math.sin(y) / y
	elif x <= 4e+153:
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * math.cosh(x)
	else:
		tmp = 0.5 * ((math.sin(y) * (x * x)) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.5e-9)
		tmp = Float64(sin(y) / y);
	elseif (x <= 4e+153)
		tmp = Float64(Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))) * cosh(x));
	else
		tmp = Float64(0.5 * Float64(Float64(sin(y) * Float64(x * x)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.5e-9)
		tmp = sin(y) / y;
	elseif (x <= 4e+153)
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * cosh(x);
	else
		tmp = 0.5 * ((sin(y) * (x * x)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.5e-9], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 4e+153], N[(N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[Sin[y], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.4999999999999999e-9

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 85.0%

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

      1. unpow2 85.0%

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

   4. Simplified 85.0%

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

      1. clear-num 85.0%

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

      2. associate-/r/ 84.9%

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

   6. Applied egg-rr 84.9%

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

      1. /-rgt-identity 84.9%

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

      2. associate-/r/ 85.0%

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

   8. Applied egg-rr 85.0%

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

   9. Taylor expanded in x around 0 69.3%

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

   if 3.4999999999999999e-9 < x < 4e153

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 75.1%

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

      1. unpow2 21.8%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   4. Simplified 75.1%

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

   if 4e153 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2} \cdot \sin y}{y}} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+153}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \cosh x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sin y \cdot \left(x \cdot x\right)}{y}\\ \end{array} \]

Alternative 4: 72.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 1.44 \cdot 10^{+150}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \cosh x\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(\frac{x \cdot x}{y} \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.5e-9)
   (/ (sin y) y)
   (if (<= x 1.44e+150)
     (* (+ 1.0 (* -0.16666666666666666 (* y y))) (cosh x))
     (* (sin y) (* (/ (* x x) y) 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.5e-9) {
		tmp = sin(y) / y;
	} else if (x <= 1.44e+150) {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * cosh(x);
	} else {
		tmp = sin(y) * (((x * x) / y) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 3.5d-9) then
        tmp = sin(y) / y
    else if (x <= 1.44d+150) then
        tmp = (1.0d0 + ((-0.16666666666666666d0) * (y * y))) * cosh(x)
    else
        tmp = sin(y) * (((x * x) / y) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 3.5e-9) {
		tmp = Math.sin(y) / y;
	} else if (x <= 1.44e+150) {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * Math.cosh(x);
	} else {
		tmp = Math.sin(y) * (((x * x) / y) * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.5e-9:
		tmp = math.sin(y) / y
	elif x <= 1.44e+150:
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * math.cosh(x)
	else:
		tmp = math.sin(y) * (((x * x) / y) * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.5e-9)
		tmp = Float64(sin(y) / y);
	elseif (x <= 1.44e+150)
		tmp = Float64(Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))) * cosh(x));
	else
		tmp = Float64(sin(y) * Float64(Float64(Float64(x * x) / y) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.5e-9)
		tmp = sin(y) / y;
	elseif (x <= 1.44e+150)
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * cosh(x);
	else
		tmp = sin(y) * (((x * x) / y) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.5e-9], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 1.44e+150], N[(N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.4999999999999999e-9

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 85.0%

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

      1. unpow2 85.0%

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

   4. Simplified 85.0%

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

      1. clear-num 85.0%

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

      2. associate-/r/ 84.9%

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

   6. Applied egg-rr 84.9%

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

      1. /-rgt-identity 84.9%

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

      2. associate-/r/ 85.0%

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

   8. Applied egg-rr 85.0%

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

   9. Taylor expanded in x around 0 69.3%

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

   if 3.4999999999999999e-9 < x < 1.43999999999999998e150

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 77.5%

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

      1. unpow2 22.5%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   4. Simplified 77.5%

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

   if 1.43999999999999998e150 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 97.3%

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

      1. unpow2 97.3%

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

   4. Simplified 97.3%

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

   5. Taylor expanded in x around inf 97.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2} \cdot \sin y}{y}} \]
   6. Step-by-step derivation

      1. unpow2 97.3%

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

      2. *-commutative 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in y around inf 97.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2} \cdot \sin y}{y}} \]
   9. Step-by-step derivation

      1. *-commutative 97.3%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \sin y}{y} \cdot 0.5} \]

      2. unpow2 97.3%

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

      3. *-commutative 97.3%

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

      4. associate-*r/ 97.3%

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

      5. associate-*l* 97.3%

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

   10. Simplified 97.3%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 1.44 \cdot 10^{+150}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \cosh x\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(\frac{x \cdot x}{y} \cdot 0.5\right)\\ \end{array} \]

Alternative 5: 68.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 3.5e-9) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.5e-9) {
		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 <= 3.5d-9) 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 <= 3.5e-9) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.5e-9:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.5e-9)
		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 <= 3.5e-9)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.5e-9], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.4999999999999999e-9

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 85.0%

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

      1. unpow2 85.0%

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

   4. Simplified 85.0%

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

      1. clear-num 85.0%

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

      2. associate-/r/ 84.9%

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

   6. Applied egg-rr 84.9%

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

      1. /-rgt-identity 84.9%

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

      2. associate-/r/ 85.0%

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

   8. Applied egg-rr 85.0%

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

   9. Taylor expanded in x around 0 69.3%

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

   if 3.4999999999999999e-9 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 80.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]

Alternative 6: 61.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{+112}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.25e+112)
   (cosh x)
   (* (+ 1.0 (* -0.16666666666666666 (* y y))) (+ 1.0 (* (* x x) 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.25e+112) {
		tmp = cosh(x);
	} else {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.25d+112) then
        tmp = cosh(x)
    else
        tmp = (1.0d0 + ((-0.16666666666666666d0) * (y * y))) * (1.0d0 + ((x * x) * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.25e+112) {
		tmp = Math.cosh(x);
	} else {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.25e+112:
		tmp = math.cosh(x)
	else:
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * 0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.25e+112)
		tmp = cosh(x);
	else
		tmp = Float64(Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))) * Float64(1.0 + Float64(Float64(x * x) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.25e+112)
		tmp = cosh(x);
	else
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.25e+112], N[Cosh[x], $MachinePrecision], N[(N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.24999999999999995e112

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 73.0%

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

   if 2.24999999999999995e112 < y

   1. Initial program 99.7%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 81.6%

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

      1. unpow2 81.6%

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

   4. Simplified 81.6%

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

   5. Taylor expanded in y around 0 26.5%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {y}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 26.5%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   7. Simplified 26.5%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{+112}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \end{array} \]

Alternative 7: 50.0% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+57}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{x \cdot x}{y} \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 9.5e+57)
   (* (+ 1.0 (* -0.16666666666666666 (* y y))) (+ 1.0 (* (* x x) 0.5)))
   (* y (* (/ (* x x) y) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9.5e+57) {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * 0.5));
	} else {
		tmp = y * (((x * x) / y) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 9.5d+57) then
        tmp = (1.0d0 + ((-0.16666666666666666d0) * (y * y))) * (1.0d0 + ((x * x) * 0.5d0))
    else
        tmp = y * (((x * x) / y) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 9.5e+57) {
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * 0.5));
	} else {
		tmp = y * (((x * x) / y) * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 9.5e+57:
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * 0.5))
	else:
		tmp = y * (((x * x) / y) * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9.5e+57)
		tmp = Float64(Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))) * Float64(1.0 + Float64(Float64(x * x) * 0.5)));
	else
		tmp = Float64(y * Float64(Float64(Float64(x * x) / y) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 9.5e+57)
		tmp = (1.0 + (-0.16666666666666666 * (y * y))) * (1.0 + ((x * x) * 0.5));
	else
		tmp = y * (((x * x) / y) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9.5e+57], N[(N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.4999999999999997e57

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 80.6%

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

      1. unpow2 80.6%

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

   4. Simplified 80.6%

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

   5. Taylor expanded in y around 0 51.8%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {y}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 51.8%

         \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \left(1 + -0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   7. Simplified 51.8%

      \[\leadsto \left(1 + 0.5 \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   if 9.4999999999999997e57 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 63.8%

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

      1. unpow2 63.8%

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

   4. Simplified 63.8%

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

   5. Taylor expanded in x around inf 63.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2} \cdot \sin y}{y}} \]
   6. Step-by-step derivation

      1. unpow2 63.8%

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

      2. *-commutative 63.8%

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

   7. Simplified 63.8%

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

   8. Taylor expanded in y around inf 63.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2} \cdot \sin y}{y}} \]
   9. Step-by-step derivation

      1. *-commutative 63.8%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \sin y}{y} \cdot 0.5} \]

      2. unpow2 63.8%

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

      3. *-commutative 63.8%

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

      4. associate-*r/ 79.2%

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

      5. associate-*l* 79.2%

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

   10. Simplified 79.2%

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

   11. Taylor expanded in y around 0 70.0%

       \[\leadsto \color{blue}{y} \cdot \left(\frac{x \cdot x}{y} \cdot 0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{+57}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{x \cdot x}{y} \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 28.3% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-90}:\\ \;\;\;\;y \cdot \left(\frac{x \cdot x}{y} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot \left(x \cdot x\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4e-90) (* y (* (/ (* x x) y) 0.5)) (* 0.5 (/ (* y (* x x)) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4e-90) {
		tmp = y * (((x * x) / y) * 0.5);
	} else {
		tmp = 0.5 * ((y * (x * x)) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4d-90) then
        tmp = y * (((x * x) / y) * 0.5d0)
    else
        tmp = 0.5d0 * ((y * (x * x)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4e-90) {
		tmp = y * (((x * x) / y) * 0.5);
	} else {
		tmp = 0.5 * ((y * (x * x)) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4e-90:
		tmp = y * (((x * x) / y) * 0.5)
	else:
		tmp = 0.5 * ((y * (x * x)) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4e-90)
		tmp = Float64(y * Float64(Float64(Float64(x * x) / y) * 0.5));
	else
		tmp = Float64(0.5 * Float64(Float64(y * Float64(x * x)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e-90)
		tmp = y * (((x * x) / y) * 0.5);
	else
		tmp = 0.5 * ((y * (x * x)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4e-90], N[(y * N[(N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999998e-90

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 73.6%

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

      1. unpow2 73.6%

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

   4. Simplified 73.6%

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

   5. Taylor expanded in x around inf 26.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2} \cdot \sin y}{y}} \]
   6. Step-by-step derivation

      1. unpow2 26.3%

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

      2. *-commutative 26.3%

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

   7. Simplified 26.3%

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

   8. Taylor expanded in y around inf 26.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2} \cdot \sin y}{y}} \]
   9. Step-by-step derivation

      1. *-commutative 26.3%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \sin y}{y} \cdot 0.5} \]

      2. unpow2 26.3%

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

      3. *-commutative 26.3%

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

      4. associate-*r/ 35.6%

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

      5. associate-*l* 35.6%

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

   10. Simplified 35.6%

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

   11. Taylor expanded in y around 0 32.0%

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

   if 3.99999999999999998e-90 < y

   1. Initial program 99.8%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 84.2%

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

      1. unpow2 84.2%

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

   4. Simplified 84.2%

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

   5. Taylor expanded in x around inf 33.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2} \cdot \sin y}{y}} \]
   6. Step-by-step derivation

      1. unpow2 33.9%

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

      2. *-commutative 33.9%

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

   7. Simplified 33.9%

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

   8. Taylor expanded in y around 0 27.1%

      \[\leadsto 0.5 \cdot \frac{\color{blue}{{x}^{2} \cdot y}}{y} \]
   9. Step-by-step derivation

      1. unpow2 27.1%

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

      2. *-commutative 27.1%

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

   10. Simplified 27.1%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{y \cdot \left(x \cdot x\right)}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-90}:\\ \;\;\;\;y \cdot \left(\frac{x \cdot x}{y} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot \left(x \cdot x\right)}{y}\\ \end{array} \]

Alternative 9: 47.7% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.45:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{x \cdot x}{y} \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.45) (+ 1.0 (* (* x x) 0.5)) (* y (* (/ (* x x) y) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.45) {
		tmp = 1.0 + ((x * x) * 0.5);
	} else {
		tmp = y * (((x * x) / y) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.45d0) then
        tmp = 1.0d0 + ((x * x) * 0.5d0)
    else
        tmp = y * (((x * x) / y) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.45) {
		tmp = 1.0 + ((x * x) * 0.5);
	} else {
		tmp = y * (((x * x) / y) * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.45:
		tmp = 1.0 + ((x * x) * 0.5)
	else:
		tmp = y * (((x * x) / y) * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.45)
		tmp = Float64(1.0 + Float64(Float64(x * x) * 0.5));
	else
		tmp = Float64(y * Float64(Float64(Float64(x * x) / y) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.45)
		tmp = 1.0 + ((x * x) * 0.5);
	else
		tmp = y * (((x * x) / y) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.45], N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4500000000000002

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in y around 0 58.5%

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

   3. Taylor expanded in x around 0 48.4%

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

      1. unpow2 85.0%

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

   5. Simplified 48.4%

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

   if 2.4500000000000002 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]

   2. Taylor expanded in x around 0 53.9%

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

      1. unpow2 53.9%

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

   4. Simplified 53.9%

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

   5. Taylor expanded in x around inf 53.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2} \cdot \sin y}{y}} \]
   6. Step-by-step derivation

      1. unpow2 53.9%

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

      2. *-commutative 53.9%

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

   7. Simplified 53.9%

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

   8. Taylor expanded in y around inf 53.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2} \cdot \sin y}{y}} \]
   9. Step-by-step derivation

      1. *-commutative 53.9%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot \sin y}{y} \cdot 0.5} \]

      2. unpow2 53.9%

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

      3. *-commutative 53.9%

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

      4. associate-*r/ 68.2%

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

      5. associate-*l* 68.2%

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

   10. Simplified 68.2%

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

   11. Taylor expanded in y around 0 60.2%

       \[\leadsto \color{blue}{y} \cdot \left(\frac{x \cdot x}{y} \cdot 0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.45:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{x \cdot x}{y} \cdot 0.5\right)\\ \end{array} \]

Alternative 10: 24.3% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]

2. Taylor expanded in x around 0 77.0%

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

   1. unpow2 77.0%

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

4. Simplified 77.0%

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

5. Taylor expanded in x around inf 28.8%

   \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2} \cdot \sin y}{y}} \]
6. Step-by-step derivation

   1. unpow2 28.8%

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

   2. *-commutative 28.8%

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

7. Simplified 28.8%

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

8. Taylor expanded in y around 0 25.2%

   \[\leadsto 0.5 \cdot \frac{\color{blue}{{x}^{2} \cdot y}}{y} \]
9. Step-by-step derivation

   1. unpow2 25.2%

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

   2. *-commutative 25.2%

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

10. Simplified 25.2%

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

11. Final simplification 25.2%

    \[\leadsto 0.5 \cdot \frac{y \cdot \left(x \cdot x\right)}{y} \]

Alternative 11: 21.9% accurate, 41.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* x (* x 0.5)))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x * (x * 0.5)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]

2. Taylor expanded in x around 0 77.0%

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

   1. unpow2 77.0%

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

4. Simplified 77.0%

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

5. Taylor expanded in x around inf 28.8%

   \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2} \cdot \sin y}{y}} \]
6. Step-by-step derivation

   1. unpow2 28.8%

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

   2. *-commutative 28.8%

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

7. Simplified 28.8%

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

8. Taylor expanded in y around 0 23.1%

   \[\leadsto \color{blue}{0.5 \cdot {x}^{2}} \]
9. Step-by-step derivation

   1. unpow2 23.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot x\right)} \]

   2. *-commutative 23.1%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 0.5} \]

   3. associate-*r* 23.1%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.5\right)} \]

10. Simplified 23.1%

    \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.5\right)} \]

11. Final simplification 23.1%

    \[\leadsto x \cdot \left(x \cdot 0.5\right) \]

Developer target: 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 2023271 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

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

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