Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 8.7s

Alternatives: 5

Speedup: 1.0×

• 50.7% 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: 86.2% 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 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.2%

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

   if 1.05000000000000004 < (cosh.f64 x)

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. associate-/l/ 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. clear-num 100.0%

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

      3. *-un-lft-identity 100.0%

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

      4. times-frac 100.0%

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

      5. clear-num 100.0%

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

      6. frac-times 100.0%

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

      7. *-un-lft-identity 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in y around 0 69.0%

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

4. Final simplification 84.0%

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

Alternative 3: 62.0% 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. Step-by-step derivation

   1. *-commutative 99.9%

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

   2. associate-/r/ 99.9%

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

3. Simplified 99.9%

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

   1. clear-num 99.9%

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

   2. inv-pow 99.9%

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

   3. associate-/l/ 99.9%

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

5. Applied egg-rr 99.9%

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

   1. unpow-1 99.9%

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

   2. clear-num 99.9%

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

   3. *-un-lft-identity 99.9%

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

   4. times-frac 99.8%

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

   5. clear-num 99.8%

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

   6. frac-times 99.8%

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

   7. *-un-lft-identity 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in y around 0 64.0%

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

9. Final simplification 64.0%

   \[\leadsto \cosh x \]

Alternative 4: 25.9% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 / N[(y * N[(N[(y * 0.16666666666666666), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $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-/r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 50.5%

   \[\leadsto \color{blue}{\frac{\sin y}{y}} \]
5. Step-by-step derivation

   1. clear-num 50.5%

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

   2. inv-pow 50.5%

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

6. Applied egg-rr 50.5%

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

   1. unpow-1 50.5%

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

8. Simplified 50.5%

   \[\leadsto \color{blue}{\frac{1}{\frac{y}{\sin y}}} \]
9. Step-by-step derivation

   1. *-un-lft-identity 50.5%

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

   2. associate-*l/ 50.4%

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

10. Applied egg-rr 50.4%

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

11. Taylor expanded in y around 0 30.6%

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

12. Final simplification 30.6%

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

Alternative 5: 25.3% 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-/r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 50.5%

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

5. Taylor expanded in y around 0 30.4%

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

6. Final simplification 30.4%

   \[\leadsto 1 \]

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 2023298 
(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)))