Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.5% → 99.8%

Time: 10.8s

Alternatives: 14

Speedup: 1.0×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{\sinh y}{\frac{x}{\sin x}}} \]
5. Add Preprocessing

Alternative 2: 72.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot x\right)}{x}\\ \mathbf{elif}\;t\_0 \leq 10^{-110}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (<= t_0 (- INFINITY))
     (/
      (*
       (*
        (fma
         (fma (* 0.008333333333333333 y) y 0.16666666666666666)
         (* y y)
         1.0)
        y)
       (* (* (* x x) -0.16666666666666666) x))
      x)
     (if (<= t_0 1e-110) (* (/ (sin x) x) y) (/ (sinh y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = ((fma(fma((0.008333333333333333 * y), y, 0.16666666666666666), (y * y), 1.0) * y) * (((x * x) * -0.16666666666666666) * x)) / x;
	} else if (t_0 <= 1e-110) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = sinh(y) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(fma(fma(Float64(0.008333333333333333 * y), y, 0.16666666666666666), Float64(y * y), 1.0) * y) * Float64(Float64(Float64(x * x) * -0.16666666666666666) * x)) / x);
	elseif (t_0 <= 1e-110)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = Float64(sinh(y) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(N[(N[(N[(0.008333333333333333 * y), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[t$95$0, 1e-110], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 57.9

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

   8. Applied rewrites 57.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\left(\frac{-1}{6} \cdot \color{blue}{{x}^{3}}\right) \cdot \left(\mathsf{fma}\left(\frac{1}{6} \cdot y, y, 1\right) \cdot y\right)}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 23.8%

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

   12. Taylor expanded in y around 0

       \[\leadsto \frac{\left(\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)}}{x} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\left(\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \color{blue}{\left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y\right)}}{x} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \color{blue}{\left(\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot y\right)}}{x} \]

       3. +-commutative N/A

          \[\leadsto \frac{\left(\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) + 1\right)} \cdot y\right)}{x} \]

       4. *-commutative N/A

          \[\leadsto \frac{\left(\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\left(\color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}} + 1\right) \cdot y\right)}{x} \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}, {y}^{2}, 1\right)} \cdot y\right)}{x} \]

       6. +-commutative N/A

          \[\leadsto \frac{\left(\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}}, {y}^{2}, 1\right) \cdot y\right)}{x} \]

       7. unpow2 N/A

          \[\leadsto \frac{\left(\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(\frac{1}{120} \cdot \color{blue}{\left(y \cdot y\right)} + \frac{1}{6}, {y}^{2}, 1\right) \cdot y\right)}{x} \]

       8. associate-*r* N/A

          \[\leadsto \frac{\left(\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} \cdot y\right) \cdot y} + \frac{1}{6}, {y}^{2}, 1\right) \cdot y\right)}{x} \]

       9. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} \cdot y, y, \frac{1}{6}\right)}, {y}^{2}, 1\right) \cdot y\right)}{x} \]

       10. lower-*.f64 N/A

           \[\leadsto \frac{\left(\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot y}, y, \frac{1}{6}\right), {y}^{2}, 1\right) \cdot y\right)}{x} \]

       11. unpow2 N/A

           \[\leadsto \frac{\left(\left(\frac{-1}{6} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{120} \cdot y, y, \frac{1}{6}\right), \color{blue}{y \cdot y}, 1\right) \cdot y\right)}{x} \]

       12. lower-*.f64 24.3

           \[\leadsto \frac{\left(\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right) \cdot y\right)}{x} \]

   14. Applied rewrites 24.3%

       \[\leadsto \frac{\left(\left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right)}}{x} \]

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.0000000000000001e-110

   1. Initial program 77.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 1.0000000000000001e-110 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 74.7%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -\infty:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333 \cdot y, y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right) \cdot \left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot x\right)}{x}\\ \mathbf{elif}\;\frac{\sin x \cdot \sinh y}{x} \leq 10^{-110}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024230 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

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

  (/ (* (sin x) (sinh y)) x))