Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 11.6s

Alternatives: 23

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 3.7519488489777915e+33
  2. y = 5.90057761534014e+37

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

   \[\leadsto \frac{\sinh y}{y} \cdot \cos x \]
4. Add Preprocessing

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := t\_0 \cdot \cos x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot y, y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(-0.5 \cdot x, x, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999999992255:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y, y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* t_0 (cos x))))
   (if (<= t_1 (- INFINITY))
     (*
      (fma
       (fma
        (fma (* 0.0001984126984126984 y) y 0.008333333333333333)
        (* y y)
        0.16666666666666666)
       (* y y)
       1.0)
      (fma (* -0.5 x) x 1.0))
     (if (<= t_1 0.9999999999992255)
       (*
        (fma
         (*
          (fma
           (* (fma 0.0001984126984126984 (* y y) 0.008333333333333333) y)
           y
           0.16666666666666666)
          y)
         y
         1.0)
        (cos x))
       (* 1.0 t_0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = t_0 * cos(x);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(fma(fma((0.0001984126984126984 * y), y, 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * fma((-0.5 * x), x, 1.0);
	} else if (t_1 <= 0.9999999999992255) {
		tmp = fma((fma((fma(0.0001984126984126984, (y * y), 0.008333333333333333) * y), y, 0.16666666666666666) * y), y, 1.0) * cos(x);
	} else {
		tmp = 1.0 * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(t_0 * cos(x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(fma(fma(Float64(0.0001984126984126984 * y), y, 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * fma(Float64(-0.5 * x), x, 1.0));
	elseif (t_1 <= 0.9999999999992255)
		tmp = Float64(fma(Float64(fma(Float64(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333) * y), y, 0.16666666666666666) * y), y, 1.0) * cos(x));
	else
		tmp = Float64(1.0 * t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(0.0001984126984126984 * y), $MachinePrecision] * y + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(-0.5 * x), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999999992255], N[(N[(N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * y), $MachinePrecision] * y + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(1.0 * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 52.9%

      \[\leadsto \cos x \cdot \left(\left(y \cdot 0.16666666666666666\right) \cdot \color{blue}{y}\right) \]

   9. Taylor expanded in y around 0

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \cos x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)} \]

       2. *-commutative N/A

          \[\leadsto \cos x \cdot \left(\color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) \cdot {y}^{2}} + 1\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \cos x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right), {y}^{2}, 1\right)} \]

       4. +-commutative N/A

          \[\leadsto \cos x \cdot \mathsf{fma}\left(\color{blue}{{y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) + \frac{1}{6}}, {y}^{2}, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \cos x \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right) \cdot {y}^{2}} + \frac{1}{6}, {y}^{2}, 1\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \cos x \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}, {y}^{2}, \frac{1}{6}\right)}, {y}^{2}, 1\right) \]

       7. +-commutative N/A

          \[\leadsto \cos x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{5040} \cdot {y}^{2} + \frac{1}{120}}, {y}^{2}, \frac{1}{6}\right), {y}^{2}, 1\right) \]

       8. unpow2 N/A

          \[\leadsto \cos x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040} \cdot \color{blue}{\left(y \cdot y\right)} + \frac{1}{120}, {y}^{2}, \frac{1}{6}\right), {y}^{2}, 1\right) \]

       9. associate-*r* N/A

          \[\leadsto \cos x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{5040} \cdot y\right) \cdot y} + \frac{1}{120}, {y}^{2}, \frac{1}{6}\right), {y}^{2}, 1\right) \]

       10. *-commutative N/A

           \[\leadsto \cos x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(y \cdot \frac{1}{5040}\right)} \cdot y + \frac{1}{120}, {y}^{2}, \frac{1}{6}\right), {y}^{2}, 1\right) \]

       11. lower-fma.f64 N/A

           \[\leadsto \cos x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y \cdot \frac{1}{5040}, y, \frac{1}{120}\right)}, {y}^{2}, \frac{1}{6}\right), {y}^{2}, 1\right) \]

       12. *-commutative N/A

           \[\leadsto \cos x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{5040} \cdot y}, y, \frac{1}{120}\right), {y}^{2}, \frac{1}{6}\right), {y}^{2}, 1\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \cos x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{5040} \cdot y}, y, \frac{1}{120}\right), {y}^{2}, \frac{1}{6}\right), {y}^{2}, 1\right) \]

       14. unpow2 N/A

           \[\leadsto \cos x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040} \cdot y, y, \frac{1}{120}\right), \color{blue}{y \cdot y}, \frac{1}{6}\right), {y}^{2}, 1\right) \]

       15. lower-*.f64 N/A

           \[\leadsto \cos x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040} \cdot y, y, \frac{1}{120}\right), \color{blue}{y \cdot y}, \frac{1}{6}\right), {y}^{2}, 1\right) \]

       16. unpow2 N/A

           \[\leadsto \cos x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040} \cdot y, y, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right), \color{blue}{y \cdot y}, 1\right) \]

       17. lower-*.f64 86.3

           \[\leadsto \cos x \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot y, y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), \color{blue}{y \cdot y}, 1\right) \]

   11. Applied rewrites 86.3%

       \[\leadsto \cos x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot y, y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)} \]

   12. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040} \cdot y, y, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right) \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot {x}^{2} + 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040} \cdot y, y, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right) \]

       2. unpow2 N/A

          \[\leadsto \left(\frac{-1}{2} \cdot \color{blue}{\left(x \cdot x\right)} + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040} \cdot y, y, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right) \]

       3. associate-*r* N/A

          \[\leadsto \left(\color{blue}{\left(\frac{-1}{2} \cdot x\right) \cdot x} + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040} \cdot y, y, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot x, x, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{5040} \cdot y, y, \frac{1}{120}\right), y \cdot y, \frac{1}{6}\right), y \cdot y, 1\right) \]

       5. lower-*.f64 96.9

          \[\leadsto \mathsf{fma}\left(\color{blue}{-0.5 \cdot x}, x, 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot y, y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \]

   14. Applied rewrites 96.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot x, x, 1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot y, y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \]

   if -inf.0 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y)) < 0.9999999999992255

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos x \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right)} \]

      2. unpow2 N/A

         \[\leadsto \cos x \cdot \left(\color{blue}{\left(y \cdot y\right)} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right) + 1\right) \]

      3. associate-*l* N/A

         \[\leadsto \cos x \cdot \left(\color{blue}{y \cdot \left(y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot \left(\color{blue}{\left(y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right) \cdot y} + 1\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \cos x \cdot \color{blue}{\mathsf{fma}\left(y \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right), y, 1\right)} \]

   5. Applied rewrites 99.6%

      \[\leadsto \cos x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y, y, 0.16666666666666666\right) \cdot y, y, 1\right)} \]

   if 0.9999999999992255 < (*.f64 (cos.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \cos x \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot y, y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \mathsf{fma}\left(-0.5 \cdot x, x, 1\right)\\ \mathbf{elif}\;\frac{\sinh y}{y} \cdot \cos x \leq 0.9999999999992255:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y, y, 0.16666666666666666\right) \cdot y, y, 1\right) \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{\sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))