Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 10.5s

Alternatives: 27

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 2.4435896263074772e+250
  2. y = 2.065451670483147e-301

• 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

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

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(Float64(sin(y) / y) * cosh(x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin y}{y} \cdot \cosh x\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right) \cdot \cosh x\\ \mathbf{elif}\;t\_0 \leq 0.9999984954924755:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot x, x, 0.5\right) \cdot x, x, 1\right) \cdot \sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ (sin y) y) (cosh x))))
   (if (<= t_0 (- INFINITY))
     (* (fma (* -0.16666666666666666 y) y 1.0) (cosh x))
     (if (<= t_0 0.9999984954924755)
       (/
        (* (fma (* (fma (* 0.041666666666666664 x) x 0.5) x) x 1.0) (sin y))
        y)
       (* 1.0 (cosh x))))))

C

double code(double x, double y) {
	double t_0 = (sin(y) / y) * cosh(x);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma((-0.16666666666666666 * y), y, 1.0) * cosh(x);
	} else if (t_0 <= 0.9999984954924755) {
		tmp = (fma((fma((0.041666666666666664 * x), x, 0.5) * x), x, 1.0) * sin(y)) / y;
	} else {
		tmp = 1.0 * cosh(x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(y) / y) * cosh(x))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(fma(Float64(-0.16666666666666666 * y), y, 1.0) * cosh(x));
	elseif (t_0 <= 0.9999984954924755)
		tmp = Float64(Float64(fma(Float64(fma(Float64(0.041666666666666664 * x), x, 0.5) * x), x, 1.0) * sin(y)) / y);
	else
		tmp = Float64(1.0 * cosh(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999984954924755], N[(N[(N[(N[(N[(N[(0.041666666666666664 * x), $MachinePrecision] * x + 0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(1.0 * N[Cosh[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < -inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cosh x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {y}^{2} + 1\right)} \]

      2. unpow2 N/A

         \[\leadsto \cosh x \cdot \left(\frac{-1}{6} \cdot \color{blue}{\left(y \cdot y\right)} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \cosh x \cdot \left(\color{blue}{\left(\frac{-1}{6} \cdot y\right) \cdot y} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \cosh x \cdot \left(\color{blue}{\left(y \cdot \frac{-1}{6}\right)} \cdot y + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \cosh x \cdot \color{blue}{\mathsf{fma}\left(y \cdot \frac{-1}{6}, y, 1\right)} \]

      6. *-commutative N/A

         \[\leadsto \cosh x \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{6} \cdot y}, y, 1\right) \]

      7. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y)) < 0.999998495492475548

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\sin y \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}}{y} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin y \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)}}{y} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin y \cdot \left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1\right)}{y} \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin y \cdot \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)} + 1\right)}{y} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\sin y \cdot \left(\color{blue}{\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot x\right) \cdot x} + 1\right)}{y} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin y \cdot \color{blue}{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot x, x, 1\right)}}{y} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin y \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot x}, x, 1\right)}{y} \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin y \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)} \cdot x, x, 1\right)}{y} \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin y \cdot \mathsf{fma}\left(\left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{2}\right) \cdot x, x, 1\right)}{y} \]

      9. associate-*r* N/A

         \[\leadsto \frac{\sin y \cdot \mathsf{fma}\left(\left(\color{blue}{\left(\frac{1}{24} \cdot x\right) \cdot x} + \frac{1}{2}\right) \cdot x, x, 1\right)}{y} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sin y \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot x, x, \frac{1}{2}\right)} \cdot x, x, 1\right)}{y} \]

      11. lower-*.f64 98.6

          \[\leadsto \frac{\sin y \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.041666666666666664 \cdot x}, x, 0.5\right) \cdot x, x, 1\right)}{y} \]

   7. Applied rewrites 98.6%

      \[\leadsto \frac{\sin y \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot x, x, 0.5\right) \cdot x, x, 1\right)}}{y} \]

   if 0.999998495492475548 < (*.f64 (cosh.f64 x) (/.f64 (sin.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cosh x \cdot \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot \cosh x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right) \cdot \cosh x\\ \mathbf{elif}\;\frac{\sin y}{y} \cdot \cosh x \leq 0.9999984954924755:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664 \cdot x, x, 0.5\right) \cdot x, x, 1\right) \cdot \sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \cosh x\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 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 2024230 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (* (cosh x) (sin y)) y))

  (* (cosh x) (/ (sin y) y)))