Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 11.3s

Alternatives: 6

Speedup: 1.0×

• 49.5% 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, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.5 \cdot \frac{\sin y \cdot \left(e^{x\_m} + \frac{1}{e^{x\_m}}\right)}{y} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (* 0.5 (/ (* (sin y) (+ (exp x_m) (/ 1.0 (exp x_m)))) y)))

C

x_m = fabs(x);
double code(double x_m, double y) {
	return 0.5 * ((sin(y) * (exp(x_m) + (1.0 / exp(x_m)))) / y);
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	return 0.5 * ((Math.sin(y) * (Math.exp(x_m) + (1.0 / Math.exp(x_m)))) / y);
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	return 0.5 * ((math.sin(y) * (math.exp(x_m) + (1.0 / math.exp(x_m)))) / y)

Julia

x_m = abs(x)
function code(x_m, y)
	return Float64(0.5 * Float64(Float64(sin(y) * Float64(exp(x_m) + Float64(1.0 / exp(x_m)))) / y))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := N[(0.5 * N[(N[(N[Sin[y], $MachinePrecision] * N[(N[Exp[x$95$m], $MachinePrecision] + N[(1.0 / N[Exp[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\cosh x\_m \leq 1.000000001:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= (cosh x_m) 1.000000001) (/ (sin y) y) (cosh x_m)))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (cosh(x_m) <= 1.000000001) {
		tmp = sin(y) / y;
	} else {
		tmp = cosh(x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: tmp
    if (cosh(x_m) <= 1.000000001d0) then
        tmp = sin(y) / y
    else
        tmp = cosh(x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double tmp;
	if (Math.cosh(x_m) <= 1.000000001) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if math.cosh(x_m) <= 1.000000001:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (cosh(x_m) <= 1.000000001)
		tmp = Float64(sin(y) / y);
	else
		tmp = cosh(x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if (cosh(x_m) <= 1.000000001)
		tmp = sin(y) / y;
	else
		tmp = cosh(x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[N[Cosh[x$95$m], $MachinePrecision], 1.000000001], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x$95$m], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.0000000010000001

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.3%

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

   if 1.0000000010000001 < (cosh.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 76.6%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.000000001:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \cosh x\_m \cdot \frac{\sin y}{y} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y) :precision binary64 (* (cosh x_m) (/ (sin y) y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := N[(N[Cosh[x$95$m], $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. Final simplification 99.9%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\sin y}{\frac{y}{\cosh x\_m}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y) :precision binary64 (/ (sin y) (/ y (cosh x_m))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	return sin(y) / (y / cosh(x_m));
}

Fortran

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

Java

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

Python

x_m = math.fabs(x)
def code(x_m, y):
	return math.sin(y) / (y / math.cosh(x_m))

Julia

x_m = abs(x)
function code(x_m, y)
	return Float64(sin(y) / Float64(y / cosh(x_m)))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := N[(N[Sin[y], $MachinePrecision] / N[(y / N[Cosh[x$95$m], $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-*l/ 99.9%

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

   3. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 48.6%

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

   2. pow2 48.6%

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

6. Applied egg-rr 48.6%

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

   1. unpow2 48.6%

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

   2. add-sqr-sqrt 99.8%

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

   3. clear-num 99.8%

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

   4. un-div-inv 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

   \[\leadsto \frac{\sin y}{\frac{y}{\cosh x}} \]
10. Add Preprocessing

Alternative 5: 63.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \cosh x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y) :precision binary64 (cosh x_m))

C

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

Fortran

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

Java

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

Python

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

Julia

x_m = abs(x)
function code(x_m, y)
	return cosh(x_m)
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := N[Cosh[x$95$m], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0 62.1%

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

4. Final simplification 62.1%

   \[\leadsto \cosh x \]
5. Add Preprocessing

Alternative 6: 26.4% accurate, 205.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y) :precision binary64 1.0)

C

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

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	return 1.0;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	return 1.0

Julia

x_m = abs(x)
function code(x_m, y)
	return 1.0
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, 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* 99.8%

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

3. Simplified 99.8%

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

   1. add-sqr-sqrt 48.6%

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

   2. pow2 48.6%

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

6. Applied egg-rr 48.6%

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

7. Taylor expanded in x around 0 25.6%

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

8. Taylor expanded in y around 0 29.4%

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

9. Final simplification 29.4%

   \[\leadsto 1 \]
10. Add Preprocessing

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 2024078 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

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

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