Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 6.8s

Alternatives: 5

Speedup: 1.0×

• 48.6% 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: 87.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.8%

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

   if 1 < (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 74.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \leq 1:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+199}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(y \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.4e+199) (cosh x) (+ 1.0 (* y (* y -0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.4e+199) {
		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 <= 4.4d+199) 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 <= 4.4e+199) {
		tmp = Math.cosh(x);
	} else {
		tmp = 1.0 + (y * (y * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.4e+199:
		tmp = math.cosh(x)
	else:
		tmp = 1.0 + (y * (y * -0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.4e+199)
		tmp = cosh(x);
	else
		tmp = Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.4e+199)
		tmp = cosh(x);
	else
		tmp = 1.0 + (y * (y * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.4e+199], N[Cosh[x], $MachinePrecision], N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.4000000000000003e199

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 67.0%

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

   if 4.4000000000000003e199 < y

   1. Initial program 99.9%

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

      1. add-log-exp 54.0%

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

      2. *-un-lft-identity 54.0%

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

      3. log-prod 54.0%

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

      4. metadata-eval 54.0%

         \[\leadsto \color{blue}{0} + \log \left(e^{\cosh x \cdot \frac{\sin y}{y}}\right) \]

      5. add-log-exp 99.9%

         \[\leadsto 0 + \color{blue}{\cosh x \cdot \frac{\sin y}{y}} \]

      6. *-commutative 99.9%

         \[\leadsto 0 + \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0 + \frac{\sin y}{y} \cdot \cosh x} \]
   5. Step-by-step derivation

      1. +-lft-identity 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* 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around 0 50.8%

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

   8. Taylor expanded in y around 0 46.2%

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

      1. unpow2 46.2%

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

      2. associate-*r* 46.2%

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

   10. Applied egg-rr 46.2%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+199}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(y \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 33.3% accurate, 29.3× speedup

Math

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

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(y * Float64(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[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. add-log-exp 74.7%

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

   2. *-un-lft-identity 74.7%

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

   3. log-prod 74.7%

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

   4. metadata-eval 74.7%

      \[\leadsto \color{blue}{0} + \log \left(e^{\cosh x \cdot \frac{\sin y}{y}}\right) \]

   5. add-log-exp 99.9%

      \[\leadsto 0 + \color{blue}{\cosh x \cdot \frac{\sin y}{y}} \]

   6. *-commutative 99.9%

      \[\leadsto 0 + \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{0 + \frac{\sin y}{y} \cdot \cosh x} \]
5. Step-by-step derivation

   1. +-lft-identity 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* 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in x around 0 50.0%

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

8. Taylor expanded in y around 0 32.1%

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

   1. unpow2 32.1%

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

   2. associate-*r* 32.1%

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

10. Applied egg-rr 32.1%

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

11. Final simplification 32.1%

    \[\leadsto 1 + y \cdot \left(y \cdot -0.16666666666666666\right) \]
12. Add Preprocessing

Alternative 5: 27.6% 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. Add Preprocessing
3. Step-by-step derivation

   1. add-log-exp 74.7%

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

   2. *-un-lft-identity 74.7%

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

   3. log-prod 74.7%

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

   4. metadata-eval 74.7%

      \[\leadsto \color{blue}{0} + \log \left(e^{\cosh x \cdot \frac{\sin y}{y}}\right) \]

   5. add-log-exp 99.9%

      \[\leadsto 0 + \color{blue}{\cosh x \cdot \frac{\sin y}{y}} \]

   6. *-commutative 99.9%

      \[\leadsto 0 + \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{0 + \frac{\sin y}{y} \cdot \cosh x} \]
5. Step-by-step derivation

   1. +-lft-identity 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* 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in x around 0 50.0%

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

8. Taylor expanded in y around 0 25.1%

   \[\leadsto \color{blue}{1} \]
9. 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 2024096 
(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)))