Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 7.2s

Alternatives: 8

Speedup: 1.0×

• 49.0% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-*l/ 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.05:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.05)
   (/ (sin y) y)
   (* (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.05) {
		tmp = sin(y) / y;
	} else {
		tmp = cosh(x) * (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 (cosh(x) <= 1.05d0) then
        tmp = sin(y) / y
    else
        tmp = cosh(x) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.cosh(x) <= 1.05) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.05:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.05)
		tmp = Float64(sin(y) / y);
	else
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (cosh(x) <= 1.05)
		tmp = sin(y) / y;
	else
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.05], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.05000000000000004

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 98.9%

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

   if 1.05000000000000004 < (cosh.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 75.8%

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

      1. unpow2 16.6%

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

   4. Simplified 75.8%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.05:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 3: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.05:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.05) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.05) {
		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 (cosh(x) <= 1.05d0) 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 (Math.cosh(x) <= 1.05) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.05:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.05)
		tmp = Float64(sin(y) / y);
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (cosh(x) <= 1.05)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.05], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.05000000000000004

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 98.9%

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

   if 1.05000000000000004 < (cosh.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 70.8%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.05:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]

Alternative 4: 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 99.9%

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

2. Final simplification 99.9%

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

Alternative 5: 84.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.02 \lor \neg \left(x \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{\left(\sin y \cdot 0.5\right) \cdot \left(2 + x \cdot x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 0.02) (not (<= x 1.35e+154)))
   (/ (* (* (sin y) 0.5) (+ 2.0 (* x x))) y)
   (* (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 0.02) || !(x <= 1.35e+154)) {
		tmp = ((sin(y) * 0.5) * (2.0 + (x * x))) / y;
	} else {
		tmp = cosh(x) * (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 ((x <= 0.02d0) .or. (.not. (x <= 1.35d+154))) then
        tmp = ((sin(y) * 0.5d0) * (2.0d0 + (x * x))) / y
    else
        tmp = cosh(x) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= 0.02) || !(x <= 1.35e+154)) {
		tmp = ((Math.sin(y) * 0.5) * (2.0 + (x * x))) / y;
	} else {
		tmp = Math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= 0.02) or not (x <= 1.35e+154):
		tmp = ((math.sin(y) * 0.5) * (2.0 + (x * x))) / y
	else:
		tmp = math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 0.02) || !(x <= 1.35e+154))
		tmp = Float64(Float64(Float64(sin(y) * 0.5) * Float64(2.0 + Float64(x * x))) / y);
	else
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= 0.02) || ~((x <= 1.35e+154)))
		tmp = ((sin(y) * 0.5) * (2.0 + (x * x))) / y;
	else
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 0.02], N[Not[LessEqual[x, 1.35e+154]], $MachinePrecision]], N[(N[(N[(N[Sin[y], $MachinePrecision] * 0.5), $MachinePrecision] * N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0200000000000000004 or 1.35000000000000003e154 < x

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l/ 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. add-log-exp 55.0%

         \[\leadsto \frac{\color{blue}{\log \left(e^{\sin y}\right)} \cdot \cosh x}{y} \]

   5. Applied egg-rr 55.0%

      \[\leadsto \frac{\color{blue}{\log \left(e^{\sin y}\right)} \cdot \cosh x}{y} \]

   6. Taylor expanded in x around 0 85.2%

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

      1. +-commutative 85.2%

         \[\leadsto \frac{\color{blue}{\sin y + 0.5 \cdot \left(\sin y \cdot {x}^{2}\right)}}{y} \]

      2. *-lft-identity 85.2%

         \[\leadsto \frac{\color{blue}{1 \cdot \sin y} + 0.5 \cdot \left(\sin y \cdot {x}^{2}\right)}{y} \]

      3. metadata-eval 85.2%

         \[\leadsto \frac{\color{blue}{\left(2 \cdot 0.5\right)} \cdot \sin y + 0.5 \cdot \left(\sin y \cdot {x}^{2}\right)}{y} \]

      4. associate-*r* 85.2%

         \[\leadsto \frac{\color{blue}{2 \cdot \left(0.5 \cdot \sin y\right)} + 0.5 \cdot \left(\sin y \cdot {x}^{2}\right)}{y} \]

      5. *-commutative 85.2%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot \sin y\right) \cdot 2} + 0.5 \cdot \left(\sin y \cdot {x}^{2}\right)}{y} \]

      6. associate-*r* 85.2%

         \[\leadsto \frac{\left(0.5 \cdot \sin y\right) \cdot 2 + \color{blue}{\left(0.5 \cdot \sin y\right) \cdot {x}^{2}}}{y} \]

      7. distribute-lft-out 85.2%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot \sin y\right) \cdot \left(2 + {x}^{2}\right)}}{y} \]

      8. *-commutative 85.2%

         \[\leadsto \frac{\color{blue}{\left(\sin y \cdot 0.5\right)} \cdot \left(2 + {x}^{2}\right)}{y} \]

      9. unpow2 85.2%

         \[\leadsto \frac{\left(\sin y \cdot 0.5\right) \cdot \left(2 + \color{blue}{x \cdot x}\right)}{y} \]

   8. Simplified 85.2%

      \[\leadsto \frac{\color{blue}{\left(\sin y \cdot 0.5\right) \cdot \left(2 + x \cdot x\right)}}{y} \]

   if 0.0200000000000000004 < x < 1.35000000000000003e154

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 77.8%

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

      1. unpow2 24.5%

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

   4. Simplified 77.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.02 \lor \neg \left(x \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{\left(\sin y \cdot 0.5\right) \cdot \left(2 + x \cdot x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 6: 62.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+247}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.7e+247) (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.7e+247) {
		tmp = cosh(x);
	} else {
		tmp = 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 <= 3.7d+247) then
        tmp = cosh(x)
    else
        tmp = 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 <= 3.7e+247) {
		tmp = Math.cosh(x);
	} else {
		tmp = 1.0 + (-0.16666666666666666 * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.7e+247:
		tmp = math.cosh(x)
	else:
		tmp = 1.0 + (-0.16666666666666666 * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.7e+247)
		tmp = cosh(x);
	else
		tmp = 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 <= 3.7e+247)
		tmp = cosh(x);
	else
		tmp = 1.0 + (-0.16666666666666666 * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.7e+247], N[Cosh[x], $MachinePrecision], N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.6999999999999998e247

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 60.0%

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

   if 3.6999999999999998e247 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 38.6%

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

   3. Taylor expanded in y around 0 25.6%

      \[\leadsto \color{blue}{1 + -0.16666666666666666 \cdot {y}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 25.6%

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

   5. Simplified 25.6%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+247}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 7: 31.9% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + -0.16666666666666666 \cdot \left(y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* -0.16666666666666666 (* y y))))

C

double code(double x, double y) {
	return 1.0 + (-0.16666666666666666 * (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((-0.16666666666666666d0) * (y * y))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (-0.16666666666666666 * (y * y));
}

Python

def code(x, y):
	return 1.0 + (-0.16666666666666666 * (y * y))

Julia

function code(x, y)
	return Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (-0.16666666666666666 * (y * y));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 50.9%

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

3. Taylor expanded in y around 0 32.1%

   \[\leadsto \color{blue}{1 + -0.16666666666666666 \cdot {y}^{2}} \]
4. Step-by-step derivation

   1. unpow2 32.1%

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

5. Simplified 32.1%

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

6. Final simplification 32.1%

   \[\leadsto 1 + -0.16666666666666666 \cdot \left(y \cdot y\right) \]

Alternative 8: 26.5% 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 99.9%

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

2. Taylor expanded in x around 0 50.9%

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

3. Taylor expanded in y around 0 25.2%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 25.2%

   \[\leadsto 1 \]

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 2023252 
(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)))