Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 8.3s

Alternatives: 9

Speedup: 1.0×

• 49.2% 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

\[\begin{array}{l} \\ \frac{\frac{\sin y}{y}}{\frac{1}{\cosh x}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (/ (sin y) y) (/ 1.0 (cosh x))))

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return (Math.sin(y) / y) / (1.0 / Math.cosh(x));
}

Python

def code(x, y):
	return (math.sin(y) / y) / (1.0 / math.cosh(x))

Julia

function code(x, y)
	return Float64(Float64(sin(y) / y) / Float64(1.0 / cosh(x)))
end

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] / N[(1.0 / N[Cosh[x], $MachinePrecision]), $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-/r/ 99.9%

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

   3. div-inv 99.9%

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

   4. associate-/r* 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \frac{\frac{\sin y}{y}}{\frac{1}{\cosh x}} \]

Alternative 2: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.01:\\ \;\;\;\;\frac{\sin y}{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.01) (/ (sin y) (+ y (* -0.5 (* y (* x x))))) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.01) {
		tmp = sin(y) / (y + (-0.5 * (y * (x * x))));
	} 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) <= 1.01d0) then
        tmp = sin(y) / (y + ((-0.5d0) * (y * (x * x))))
    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) <= 1.01) {
		tmp = Math.sin(y) / (y + (-0.5 * (y * (x * x))));
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.01:
		tmp = math.sin(y) / (y + (-0.5 * (y * (x * x))))
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.01)
		tmp = Float64(sin(y) / Float64(y + Float64(-0.5 * Float64(y * Float64(x * x)))));
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (cosh(x) <= 1.01)
		tmp = sin(y) / (y + (-0.5 * (y * (x * x))));
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.01], N[(N[Sin[y], $MachinePrecision] / N[(y + N[(-0.5 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.01000000000000001

   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-/r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.4%

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

      1. *-commutative 99.4%

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

      2. unpow2 99.4%

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

   6. Simplified 99.4%

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

   if 1.01000000000000001 < (cosh.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 72.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.01:\\ \;\;\;\;\frac{\sin y}{y + -0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]

Alternative 3: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.01:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.01) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.01) {
		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) <= 1.01d0) 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) <= 1.01) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.01:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.01)
		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) <= 1.01)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.01], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.01000000000000001

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 98.3%

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

   if 1.01000000000000001 < (cosh.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 72.6%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.01:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]

Alternative 4: 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 5: 63.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+184} \lor \neg \left(y \leq 7.5 \cdot 10^{+238}\right) \land y \leq 1.5 \cdot 10^{+262}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 4.5e+184) (and (not (<= y 7.5e+238)) (<= y 1.5e+262)))
   (cosh x)
   (* -0.16666666666666666 (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 4.5e+184) || (!(y <= 7.5e+238) && (y <= 1.5e+262))) {
		tmp = cosh(x);
	} else {
		tmp = -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 <= 4.5d+184) .or. (.not. (y <= 7.5d+238)) .and. (y <= 1.5d+262)) then
        tmp = cosh(x)
    else
        tmp = (-0.16666666666666666d0) * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 4.5e+184) || (!(y <= 7.5e+238) && (y <= 1.5e+262))) {
		tmp = Math.cosh(x);
	} else {
		tmp = -0.16666666666666666 * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 4.5e+184) or (not (y <= 7.5e+238) and (y <= 1.5e+262)):
		tmp = math.cosh(x)
	else:
		tmp = -0.16666666666666666 * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 4.5e+184) || (!(y <= 7.5e+238) && (y <= 1.5e+262)))
		tmp = cosh(x);
	else
		tmp = Float64(-0.16666666666666666 * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 4.5e+184) || (~((y <= 7.5e+238)) && (y <= 1.5e+262)))
		tmp = cosh(x);
	else
		tmp = -0.16666666666666666 * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 4.5e+184], And[N[Not[LessEqual[y, 7.5e+238]], $MachinePrecision], LessEqual[y, 1.5e+262]]], N[Cosh[x], $MachinePrecision], N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.50000000000000036e184 or 7.4999999999999996e238 < y < 1.5e262

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 69.2%

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

   if 4.50000000000000036e184 < y < 7.4999999999999996e238 or 1.5e262 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 40.7%

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

      1. unpow2 40.7%

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

   4. Simplified 40.7%

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

   5. Taylor expanded in x around 0 40.7%

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

      1. +-commutative 40.7%

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

      2. unpow2 40.7%

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

      3. fma-udef 40.7%

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

   7. Simplified 40.7%

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

   8. Taylor expanded in y around inf 40.7%

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

      1. unpow2 40.7%

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

   10. Simplified 40.7%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+184} \lor \neg \left(y \leq 7.5 \cdot 10^{+238}\right) \land y \leq 1.5 \cdot 10^{+262}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 6: 31.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{if}\;y \leq 6.8 \cdot 10^{+118}:\\ \;\;\;\;\frac{t_0 \cdot t_0 + -1}{t_0 + -1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* -0.16666666666666666 (* y y))))
   (if (<= y 6.8e+118) (/ (+ (* t_0 t_0) -1.0) (+ t_0 -1.0)) t_0)))

C

double code(double x, double y) {
	double t_0 = -0.16666666666666666 * (y * y);
	double tmp;
	if (y <= 6.8e+118) {
		tmp = ((t_0 * t_0) + -1.0) / (t_0 + -1.0);
	} else {
		tmp = t_0;
	}
	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 = (-0.16666666666666666d0) * (y * y)
    if (y <= 6.8d+118) then
        tmp = ((t_0 * t_0) + (-1.0d0)) / (t_0 + (-1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -0.16666666666666666 * (y * y);
	double tmp;
	if (y <= 6.8e+118) {
		tmp = ((t_0 * t_0) + -1.0) / (t_0 + -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -0.16666666666666666 * (y * y)
	tmp = 0
	if y <= 6.8e+118:
		tmp = ((t_0 * t_0) + -1.0) / (t_0 + -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-0.16666666666666666 * Float64(y * y))
	tmp = 0.0
	if (y <= 6.8e+118)
		tmp = Float64(Float64(Float64(t_0 * t_0) + -1.0) / Float64(t_0 + -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -0.16666666666666666 * (y * y);
	tmp = 0.0;
	if (y <= 6.8e+118)
		tmp = ((t_0 * t_0) + -1.0) / (t_0 + -1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.8e+118], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision] / N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if y < 6.79999999999999973e118

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 71.8%

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

      1. unpow2 71.8%

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

   4. Simplified 71.8%

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

   5. Taylor expanded in x around 0 31.8%

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

      1. +-commutative 31.8%

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

      2. unpow2 31.8%

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

      3. fma-udef 31.8%

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

   7. Simplified 31.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)} \]
   8. Step-by-step derivation

      1. fma-udef 31.8%

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

      2. flip-+ 28.2%

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

      3. metadata-eval 28.2%

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

   9. Applied egg-rr 28.2%

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

   if 6.79999999999999973e118 < y

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 26.3%

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

      1. unpow2 26.3%

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

   4. Simplified 26.3%

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

   5. Taylor expanded in x around 0 26.3%

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

      1. +-commutative 26.3%

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

      2. unpow2 26.3%

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

      3. fma-udef 26.3%

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

   7. Simplified 26.3%

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

   8. Taylor expanded in y around inf 26.3%

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

      1. unpow2 26.3%

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

   10. Simplified 26.3%

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

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+118}:\\ \;\;\;\;\frac{\left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right) + -1}{-0.16666666666666666 \cdot \left(y \cdot y\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 7: 29.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.98 \cdot 10^{+149}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.98e+149) 1.0 (* -0.16666666666666666 (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.98e+149) {
		tmp = 1.0;
	} else {
		tmp = -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 <= 1.98d+149) then
        tmp = 1.0d0
    else
        tmp = (-0.16666666666666666d0) * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.98e+149) {
		tmp = 1.0;
	} else {
		tmp = -0.16666666666666666 * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.98e+149:
		tmp = 1.0
	else:
		tmp = -0.16666666666666666 * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.98e+149)
		tmp = 1.0;
	else
		tmp = Float64(-0.16666666666666666 * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.98e+149)
		tmp = 1.0;
	else
		tmp = -0.16666666666666666 * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.98e+149], 1.0, N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.97999999999999995e149

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 68.9%

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

      1. unpow2 68.9%

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

   4. Simplified 68.9%

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

   5. Taylor expanded in x around 0 30.5%

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

      1. +-commutative 30.5%

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

      2. unpow2 30.5%

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

      3. fma-udef 30.5%

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

   7. Simplified 30.5%

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

   8. Taylor expanded in y around 0 25.2%

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

   if 1.97999999999999995e149 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 33.2%

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

      1. unpow2 33.2%

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

   4. Simplified 33.2%

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

   5. Taylor expanded in x around 0 33.2%

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

      1. +-commutative 33.2%

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

      2. unpow2 33.2%

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

      3. fma-udef 33.2%

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

   7. Simplified 33.2%

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

   8. Taylor expanded in y around inf 33.2%

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

      1. unpow2 33.2%

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

   10. Simplified 33.2%

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

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.98 \cdot 10^{+149}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 8: 32.6% 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 y around 0 64.1%

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

   1. unpow2 64.1%

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

4. Simplified 64.1%

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

5. Taylor expanded in x around 0 30.8%

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

   1. +-commutative 30.8%

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

   2. unpow2 30.8%

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

   3. fma-udef 30.8%

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

7. Simplified 30.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)} \]
8. Step-by-step derivation

   1. fma-udef 30.8%

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

9. Applied egg-rr 30.8%

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

10. Final simplification 30.8%

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

Alternative 9: 27.1% 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 y around 0 64.1%

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

   1. unpow2 64.1%

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

4. Simplified 64.1%

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

5. Taylor expanded in x around 0 30.8%

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

   1. +-commutative 30.8%

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

   2. unpow2 30.8%

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

   3. fma-udef 30.8%

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

7. Simplified 30.8%

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

8. Taylor expanded in y around 0 22.1%

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

9. Final simplification 22.1%

   \[\leadsto 1 \]

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 2023291 
(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)))