Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 8.0s

Alternatives: 4

Speedup: 1.0×

• 50.1% 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. Final simplification 99.9%

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

Alternative 2: 86.5% 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. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. add-sqr-sqrt 56.2%

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

      3. associate-/r* 56.2%

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

      4. *-commutative 56.2%

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

   4. Applied egg-rr 56.2%

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

   5. Taylor expanded in y around 0 78.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{x} + \frac{1}{e^{x}}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 78.1%

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

      2. metadata-eval 78.1%

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

      3. associate-/l* 78.1%

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

      4. *-rgt-identity 78.1%

         \[\leadsto \frac{\color{blue}{e^{x} + \frac{1}{e^{x}}}}{2} \]

      5. rec-exp 78.1%

         \[\leadsto \frac{e^{x} + \color{blue}{e^{-x}}}{2} \]

      6. cosh-def 78.1%

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

   7. Simplified 78.1%

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

4. Final simplification 88.2%

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

Alternative 3: 63.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cosh x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (cosh x))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return cosh(x)
end

MATLAB

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

Wolfram

code[x_, y_] := N[Cosh[x], $MachinePrecision]

Derivation

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%

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

   2. add-sqr-sqrt 51.0%

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

   3. associate-/r* 51.0%

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

   4. *-commutative 51.0%

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

4. Applied egg-rr 51.0%

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

5. Taylor expanded in y around 0 64.4%

   \[\leadsto \color{blue}{0.5 \cdot \left(e^{x} + \frac{1}{e^{x}}\right)} \]
6. Step-by-step derivation

   1. *-commutative 64.4%

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

   2. metadata-eval 64.4%

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

   3. associate-/l* 64.4%

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

   4. *-rgt-identity 64.4%

      \[\leadsto \frac{\color{blue}{e^{x} + \frac{1}{e^{x}}}}{2} \]

   5. rec-exp 64.3%

      \[\leadsto \frac{e^{x} + \color{blue}{e^{-x}}}{2} \]

   6. cosh-def 64.4%

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

7. Simplified 64.4%

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

8. Final simplification 64.4%

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

Alternative 4: 26.6% 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. 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. Taylor expanded in x around 0 48.2%

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

6. Taylor expanded in y around 0 24.4%

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

7. Final simplification 24.4%

   \[\leadsto 1 \]
8. 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 2024039 
(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)))