Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 5.2s

Alternatives: 5

Speedup: 1.0×

• 49.6% 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. Final simplification 99.9%

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

Alternative 2: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.05:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.05) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.05) {
		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) <= 1.05d0) 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) <= 1.05) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.05:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.05)
		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) <= 1.05)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.05], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.05000000000000004

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 98.8%

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

   if 1.05000000000000004 < (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 73.2%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.05:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.6% 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. Taylor expanded in y around 0 59.1%

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

4. Final simplification 59.1%

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

Alternative 4: 29.1% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+162}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{1}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.3e+162) 1.0 (/ y (/ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.3e+162) {
		tmp = 1.0;
	} else {
		tmp = y / (1.0 / 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 <= 3.3d+162) then
        tmp = 1.0d0
    else
        tmp = y / (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.3e+162) {
		tmp = 1.0;
	} else {
		tmp = y / (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.3e+162:
		tmp = 1.0
	else:
		tmp = y / (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.3e+162)
		tmp = 1.0;
	else
		tmp = Float64(y / Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.3e+162)
		tmp = 1.0;
	else
		tmp = y / (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.3e+162], 1.0, N[(y / N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.29999999999999987e162

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 53.2%

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

   4. Taylor expanded in y around 0 29.4%

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

   if 3.29999999999999987e162 < y

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 48.6%

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

      2. pow2 48.6%

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

      3. *-commutative 48.6%

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

   4. Applied egg-rr 48.6%

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

   5. Taylor expanded in x around 0 19.9%

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

      1. unpow2 19.9%

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

      2. add-sqr-sqrt 49.6%

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

      3. clear-num 47.6%

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

      4. div-inv 47.6%

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

      5. associate-/r* 49.4%

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

      6. add-exp-log 44.4%

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

      7. neg-log 44.4%

         \[\leadsto \frac{e^{\color{blue}{-\log y}}}{\frac{1}{\sin y}} \]

      8. add-sqr-sqrt 0.0%

         \[\leadsto \frac{e^{\color{blue}{\sqrt{-\log y} \cdot \sqrt{-\log y}}}}{\frac{1}{\sin y}} \]

      9. sqrt-unprod 4.7%

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

      10. sqr-neg 4.7%

          \[\leadsto \frac{e^{\sqrt{\color{blue}{\log y \cdot \log y}}}}{\frac{1}{\sin y}} \]

      11. sqrt-unprod 4.7%

          \[\leadsto \frac{e^{\color{blue}{\sqrt{\log y} \cdot \sqrt{\log y}}}}{\frac{1}{\sin y}} \]

      12. add-sqr-sqrt 4.7%

          \[\leadsto \frac{e^{\color{blue}{\log y}}}{\frac{1}{\sin y}} \]

      13. add-exp-log 4.7%

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

   7. Applied egg-rr 4.7%

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

   8. Taylor expanded in y around 0 30.0%

      \[\leadsto \frac{y}{\color{blue}{\frac{1}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+162}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{1}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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. Add Preprocessing

3. Taylor expanded in x around 0 52.6%

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

4. Taylor expanded in y around 0 25.1%

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

5. Final simplification 25.1%

   \[\leadsto 1 \]
6. 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 2024010 
(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)))