Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 5.5s

Alternatives: 5

Speedup: 1.0×

• 49.4% 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. Final simplification 99.9%

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

Alternative 2: 63.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0 64.4%

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

   1. unpow2 64.4%

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

4. Simplified 64.4%

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

5. Final simplification 64.4%

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

Alternative 3: 62.4% 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. Taylor expanded in y around 0 64.4%

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

3. Final simplification 64.4%

   \[\leadsto \cosh x \]

Alternative 4: 27.2% accurate, 18.6× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return 1.0 / (y * ((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 * ((1.0d0 / y) + (y * 0.16666666666666666d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 / N[(y * N[(N[(1.0 / y), $MachinePrecision] + N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. clear-num 99.8%

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

   2. inv-pow 99.8%

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

3. Applied egg-rr 99.8%

   \[\leadsto \cosh x \cdot \color{blue}{{\left(\frac{y}{\sin y}\right)}^{-1}} \]
4. Step-by-step derivation

   1. unpow-1 99.8%

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

   2. div-inv 99.8%

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

   3. associate-/r* 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in y around 0 58.7%

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

   1. *-commutative 58.7%

      \[\leadsto \cosh x \cdot \frac{\frac{1}{y}}{\frac{1}{y} + \color{blue}{y \cdot 0.16666666666666666}} \]

8. Simplified 58.7%

   \[\leadsto \cosh x \cdot \frac{\frac{1}{y}}{\color{blue}{\frac{1}{y} + y \cdot 0.16666666666666666}} \]

9. Taylor expanded in x around 0 29.7%

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

10. Final simplification 29.7%

    \[\leadsto \frac{1}{y \cdot \left(\frac{1}{y} + y \cdot 0.16666666666666666\right)} \]

Alternative 5: 3.0% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 99.9%

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

   1. expm1-log1p-u 99.9%

      \[\leadsto \cosh x \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin y\right)\right)}}{y} \]

   2. expm1-udef 51.9%

      \[\leadsto \cosh x \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\sin y\right)} - 1}}{y} \]

   3. log1p-udef 51.9%

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

   4. add-exp-log 51.9%

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

3. Applied egg-rr 51.9%

   \[\leadsto \cosh x \cdot \frac{\color{blue}{\left(1 + \sin y\right) - 1}}{y} \]
4. Step-by-step derivation

   1. associate--l+ 51.8%

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

5. Simplified 51.8%

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

6. Taylor expanded in y around 0 2.2%

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

7. Taylor expanded in x around 0 3.0%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 3.0%

   \[\leadsto 0 \]

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 2023214 
(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)))