Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 9.3s

Alternatives: 8

Speedup: 1.0×

• 49.3% 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 99.9%

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

Alternative 2: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.02:\\ \;\;\;\;\frac{1}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.02) (/ 1.0 (/ y (sin y))) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.02) {
		tmp = 1.0 / (y / sin(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) <= 1.02d0) then
        tmp = 1.0d0 / (y / sin(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) <= 1.02) {
		tmp = 1.0 / (y / Math.sin(y));
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.02:
		tmp = 1.0 / (y / math.sin(y))
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.02)
		tmp = Float64(1.0 / Float64(y / sin(y)));
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (cosh(x) <= 1.02)
		tmp = 1.0 / (y / sin(y));
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.02], N[(1.0 / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.02

   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* 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 98.9%

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

      1. div-inv 99.2%

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

      2. clear-num 99.2%

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

   7. Applied egg-rr 99.2%

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

   if 1.02 < (cosh.f64 x)

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 71.0%

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

      1. associate-/l* 71.0%

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

      2. *-inverses 71.0%

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

   7. Applied egg-rr 71.0%

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

      1. *-rgt-identity 71.0%

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

   9. Simplified 71.0%

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

Alternative 3: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.02:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.02) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.02) {
		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) <= 1.02d0) 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) <= 1.02) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.02:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.02)
		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) <= 1.02)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.02], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.02

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.2%

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

   if 1.02 < (cosh.f64 x)

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 71.0%

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

      1. associate-/l* 71.0%

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

      2. *-inverses 71.0%

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

   7. Applied egg-rr 71.0%

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

      1. *-rgt-identity 71.0%

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

   9. Simplified 71.0%

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

Alternative 4: 62.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+230}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5e+230) (cosh x) (+ 1.0 (* (* y y) -0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e+230) {
		tmp = cosh(x);
	} else {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5d+230) then
        tmp = cosh(x)
    else
        tmp = 1.0d0 + ((y * y) * (-0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5e+230) {
		tmp = Math.cosh(x);
	} else {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5e+230:
		tmp = math.cosh(x)
	else:
		tmp = 1.0 + ((y * y) * -0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e+230)
		tmp = cosh(x);
	else
		tmp = Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e+230)
		tmp = cosh(x);
	else
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e+230], N[Cosh[x], $MachinePrecision], N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000003e230

   1. Initial program 99.9%

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

      1. associate-*r/ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 66.8%

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

      1. associate-/l* 66.8%

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

      2. *-inverses 66.8%

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

   7. Applied egg-rr 66.8%

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

      1. *-rgt-identity 66.8%

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

   9. Simplified 66.8%

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

   if 5.0000000000000003e230 < y

   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 57.6%

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

   6. Taylor expanded in y around 0 29.5%

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

      1. *-commutative 29.5%

         \[\leadsto 1 + \color{blue}{{y}^{2} \cdot -0.16666666666666666} \]

   8. Simplified 29.5%

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

      1. unpow2 29.5%

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

   10. Applied egg-rr 29.5%

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

Alternative 5: 34.8% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + y \cdot \left(y \cdot \left(0.3333333333333333 + y \cdot -0.25\right) - 0.5\right)\\ \mathbf{if}\;x \leq 8.2 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+121}:\\ \;\;\;\;\frac{y \cdot \left(1 + y \cdot \left(y \cdot 0.3333333333333333 - 0.5\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot t\_0}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* y (- (* y (+ 0.3333333333333333 (* y -0.25))) 0.5)))))
   (if (<= x 8.2e+89)
     t_0
     (if (<= x 3.2e+121)
       (/ (* y (+ 1.0 (* y (- (* y 0.3333333333333333) 0.5)))) y)
       (/ (* y t_0) y)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y * ((y * (0.3333333333333333 + (y * -0.25))) - 0.5));
	double tmp;
	if (x <= 8.2e+89) {
		tmp = t_0;
	} else if (x <= 3.2e+121) {
		tmp = (y * (1.0 + (y * ((y * 0.3333333333333333) - 0.5)))) / y;
	} else {
		tmp = (y * t_0) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (y * ((y * (0.3333333333333333d0 + (y * (-0.25d0)))) - 0.5d0))
    if (x <= 8.2d+89) then
        tmp = t_0
    else if (x <= 3.2d+121) then
        tmp = (y * (1.0d0 + (y * ((y * 0.3333333333333333d0) - 0.5d0)))) / y
    else
        tmp = (y * t_0) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (y * ((y * (0.3333333333333333 + (y * -0.25))) - 0.5));
	double tmp;
	if (x <= 8.2e+89) {
		tmp = t_0;
	} else if (x <= 3.2e+121) {
		tmp = (y * (1.0 + (y * ((y * 0.3333333333333333) - 0.5)))) / y;
	} else {
		tmp = (y * t_0) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (y * ((y * (0.3333333333333333 + (y * -0.25))) - 0.5))
	tmp = 0
	if x <= 8.2e+89:
		tmp = t_0
	elif x <= 3.2e+121:
		tmp = (y * (1.0 + (y * ((y * 0.3333333333333333) - 0.5)))) / y
	else:
		tmp = (y * t_0) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(y * Float64(Float64(y * Float64(0.3333333333333333 + Float64(y * -0.25))) - 0.5)))
	tmp = 0.0
	if (x <= 8.2e+89)
		tmp = t_0;
	elseif (x <= 3.2e+121)
		tmp = Float64(Float64(y * Float64(1.0 + Float64(y * Float64(Float64(y * 0.3333333333333333) - 0.5)))) / y);
	else
		tmp = Float64(Float64(y * t_0) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (y * ((y * (0.3333333333333333 + (y * -0.25))) - 0.5));
	tmp = 0.0;
	if (x <= 8.2e+89)
		tmp = t_0;
	elseif (x <= 3.2e+121)
		tmp = (y * (1.0 + (y * ((y * 0.3333333333333333) - 0.5)))) / y;
	else
		tmp = (y * t_0) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y * N[(N[(y * N[(0.3333333333333333 + N[(y * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 8.2e+89], t$95$0, If[LessEqual[x, 3.2e+121], N[(N[(y * N[(1.0 + N[(y * N[(N[(y * 0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(y * t$95$0), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.1999999999999997e89

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 64.7%

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

      1. add-log-exp 31.9%

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

   5. Applied egg-rr 31.9%

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

   6. Taylor expanded in y around 0 3.6%

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

   7. Taylor expanded in y around 0 42.3%

      \[\leadsto \color{blue}{1 + y \cdot \left(y \cdot \left(0.3333333333333333 + -0.25 \cdot y\right) - 0.5\right)} \]

   if 8.1999999999999997e89 < x < 3.1999999999999999e121

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 2.3%

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

      1. add-log-exp 1.8%

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

   5. Applied egg-rr 1.8%

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

   6. Taylor expanded in y around 0 1.3%

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

   7. Taylor expanded in y around 0 59.4%

      \[\leadsto \frac{\color{blue}{y \cdot \left(1 + y \cdot \left(0.3333333333333333 \cdot y - 0.5\right)\right)}}{y} \]

   if 3.1999999999999999e121 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 2.7%

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

      1. add-log-exp 2.1%

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

   5. Applied egg-rr 2.1%

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

   6. Taylor expanded in y around 0 1.2%

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

   7. Taylor expanded in y around 0 30.5%

      \[\leadsto \frac{\color{blue}{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.3333333333333333 + -0.25 \cdot y\right) - 0.5\right)\right)}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{+89}:\\ \;\;\;\;1 + y \cdot \left(y \cdot \left(0.3333333333333333 + y \cdot -0.25\right) - 0.5\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+121}:\\ \;\;\;\;\frac{y \cdot \left(1 + y \cdot \left(y \cdot 0.3333333333333333 - 0.5\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(1 + y \cdot \left(y \cdot \left(0.3333333333333333 + y \cdot -0.25\right) - 0.5\right)\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.6% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ 1 + y \cdot \left(y \cdot \left(0.3333333333333333 + y \cdot -0.25\right) - 0.5\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ 1.0 (* y (- (* y (+ 0.3333333333333333 (* y -0.25))) 0.5))))

C

double code(double x, double y) {
	return 1.0 + (y * ((y * (0.3333333333333333 + (y * -0.25))) - 0.5));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (y * ((y * (0.3333333333333333d0 + (y * (-0.25d0)))) - 0.5d0))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (y * ((y * (0.3333333333333333 + (y * -0.25))) - 0.5));
}

Python

def code(x, y):
	return 1.0 + (y * ((y * (0.3333333333333333 + (y * -0.25))) - 0.5))

Julia

function code(x, y)
	return Float64(1.0 + Float64(y * Float64(Float64(y * Float64(0.3333333333333333 + Float64(y * -0.25))) - 0.5)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (y * ((y * (0.3333333333333333 + (y * -0.25))) - 0.5));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(y * N[(N[(y * N[(0.3333333333333333 + N[(y * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 52.6%

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

   1. add-log-exp 26.0%

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

5. Applied egg-rr 26.0%

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

6. Taylor expanded in y around 0 3.2%

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

7. Taylor expanded in y around 0 39.8%

   \[\leadsto \color{blue}{1 + y \cdot \left(y \cdot \left(0.3333333333333333 + -0.25 \cdot y\right) - 0.5\right)} \]

8. Final simplification 39.8%

   \[\leadsto 1 + y \cdot \left(y \cdot \left(0.3333333333333333 + y \cdot -0.25\right) - 0.5\right) \]
9. Add Preprocessing

Alternative 7: 32.8% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $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. 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 52.4%

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

6. Taylor expanded in y around 0 37.3%

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

   1. *-commutative 37.3%

      \[\leadsto 1 + \color{blue}{{y}^{2} \cdot -0.16666666666666666} \]

8. Simplified 37.3%

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

   1. unpow2 37.3%

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

10. Applied egg-rr 37.3%

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

Alternative 8: 26.9% 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. 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. 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 52.4%

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

6. Taylor expanded in y around 0 29.5%

   \[\leadsto \color{blue}{1} \]
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 2024182 
(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)))