Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 9.5s

Alternatives: 6

Speedup: 1.0×

• 48.9% 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

\[\begin{array}{l} \\ \frac{\sin y}{\frac{y}{\cosh x}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return Float64(sin(y) / Float64(y / cosh(x)))
end

MATLAB

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

Wolfram

code[x_, y_] := N[(N[Sin[y], $MachinePrecision] / N[(y / N[Cosh[x], $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. Final simplification 99.9%

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

Alternative 2: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.0000000002:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \cosh x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.0000000002) (/ (sin y) y) (/ (* y (cosh x)) y)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.0000000002) {
		tmp = sin(y) / y;
	} else {
		tmp = (y * cosh(x)) / y;
	}
	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) <= 1.0000000002d0) then
        tmp = sin(y) / y
    else
        tmp = (y * cosh(x)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.cosh(x) <= 1.0000000002) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = (y * Math.cosh(x)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.0000000002:
		tmp = math.sin(y) / y
	else:
		tmp = (y * math.cosh(x)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.0000000002)
		tmp = Float64(sin(y) / y);
	else
		tmp = Float64(Float64(y * cosh(x)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (cosh(x) <= 1.0000000002)
		tmp = sin(y) / y;
	else
		tmp = (y * cosh(x)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.0000000002], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[(N[(y * N[Cosh[x], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.0000000002

   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 99.8%

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

   if 1.0000000002 < (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%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in y around 0 70.5%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.0000000002:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \cosh x}{y}\\ \end{array} \]

Alternative 3: 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 4: 53.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7500000:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;{y}^{2} \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7500000.0) (/ (sin y) y) (* (pow y 2.0) -0.16666666666666666)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7500000.0) {
		tmp = sin(y) / y;
	} else {
		tmp = pow(y, 2.0) * -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 (x <= 7500000.0d0) then
        tmp = sin(y) / y
    else
        tmp = (y ** 2.0d0) * (-0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 7500000.0) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.pow(y, 2.0) * -0.16666666666666666;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7500000.0:
		tmp = math.sin(y) / y
	else:
		tmp = math.pow(y, 2.0) * -0.16666666666666666
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7500000.0)
		tmp = Float64(sin(y) / y);
	else
		tmp = Float64((y ^ 2.0) * -0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 7500000.0)
		tmp = sin(y) / y;
	else
		tmp = (y ^ 2.0) * -0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7500000.0], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[(N[Power[y, 2.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5e6

   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 70.8%

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

   if 7.5e6 < 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. Taylor expanded in x around 0 2.8%

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

      1. clear-num 2.8%

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

      2. associate-/r/ 2.8%

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

   6. Applied egg-rr 2.8%

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

   7. Taylor expanded in y around 0 22.5%

      \[\leadsto \frac{1}{y} \cdot \color{blue}{\left(y + -0.16666666666666666 \cdot {y}^{3}\right)} \]

   8. Taylor expanded in y around inf 13.0%

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

      1. *-commutative 13.0%

         \[\leadsto \color{blue}{{y}^{2} \cdot -0.16666666666666666} \]

   10. Simplified 13.0%

       \[\leadsto \color{blue}{{y}^{2} \cdot -0.16666666666666666} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7500000:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;{y}^{2} \cdot -0.16666666666666666\\ \end{array} \]

Alternative 5: 51.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (sin y) y))

C

double code(double x, double y) {
	return sin(y) / y;
}

Fortran

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

Java

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

Python

def code(x, y):
	return math.sin(y) / y

Julia

function code(x, y)
	return Float64(sin(y) / y)
end

MATLAB

function tmp = code(x, y)
	tmp = sin(y) / y;
end

Wolfram

code[x_, y_] := N[(N[Sin[y], $MachinePrecision] / y), $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 53.8%

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

5. Final simplification 53.8%

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

Alternative 6: 27.2% 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 53.8%

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

5. Taylor expanded in y around 0 31.4%

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

6. Final simplification 31.4%

   \[\leadsto 1 \]

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 2023308 
(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)))