Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 12.2s

Alternatives: 27

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 75.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ t_1 := t\_0 \cdot \sin x\\ t_2 := \mathsf{fma}\left(0.0001984126984126984 \cdot y, y, 0.008333333333333333\right)\\ t_3 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(t\_3 \cdot -0.16666666666666666\right) \cdot t\_0\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), t\_2 \cdot t\_2, -0.027777777777777776\right) \cdot y}{\mathsf{fma}\left(t\_2, y \cdot y, -0.16666666666666666\right)}, y, 1\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_3, \mathsf{fma}\left(0.008333333333333333 \cdot x, x, -0.16666666666666666\right), x\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))
        (t_1 (* t_0 (sin x)))
        (t_2 (fma (* 0.0001984126984126984 y) y 0.008333333333333333))
        (t_3 (* (* x x) x)))
   (if (<= t_1 (- INFINITY))
     (* (* t_3 -0.16666666666666666) t_0)
     (if (<= t_1 1.0)
       (*
        (fma
         (/
          (* (fma (* (* y y) (* y y)) (* t_2 t_2) -0.027777777777777776) y)
          (fma t_2 (* y y) -0.16666666666666666))
         y
         1.0)
        (sin x))
       (*
        (fma t_3 (fma (* 0.008333333333333333 x) x -0.16666666666666666) x)
        t_0)))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = t_0 * sin(x);
	double t_2 = fma((0.0001984126984126984 * y), y, 0.008333333333333333);
	double t_3 = (x * x) * x;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (t_3 * -0.16666666666666666) * t_0;
	} else if (t_1 <= 1.0) {
		tmp = fma(((fma(((y * y) * (y * y)), (t_2 * t_2), -0.027777777777777776) * y) / fma(t_2, (y * y), -0.16666666666666666)), y, 1.0) * sin(x);
	} else {
		tmp = fma(t_3, fma((0.008333333333333333 * x), x, -0.16666666666666666), x) * t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(t_0 * sin(x))
	t_2 = fma(Float64(0.0001984126984126984 * y), y, 0.008333333333333333)
	t_3 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(t_3 * -0.16666666666666666) * t_0);
	elseif (t_1 <= 1.0)
		tmp = Float64(fma(Float64(Float64(fma(Float64(Float64(y * y) * Float64(y * y)), Float64(t_2 * t_2), -0.027777777777777776) * y) / fma(t_2, Float64(y * y), -0.16666666666666666)), y, 1.0) * sin(x));
	else
		tmp = Float64(fma(t_3, fma(Float64(0.008333333333333333 * x), x, -0.16666666666666666), x) * 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[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.0001984126984126984 * y), $MachinePrecision] * y + 0.008333333333333333), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(t$95$3 * -0.16666666666666666), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] * y), $MachinePrecision] / N[(t$95$2 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 * N[(N[(0.008333333333333333 * x), $MachinePrecision] * x + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 77.0

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 26.3%

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

   if -inf.0 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.3%

      \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right) \cdot y, y, 0.16666666666666666\right) \cdot y, y, 1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.3%

      \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(0.0001984126984126984 \cdot y, y, 0.008333333333333333\right) \cdot \mathsf{fma}\left(0.0001984126984126984 \cdot y, y, 0.008333333333333333\right), -0.027777777777777776\right) \cdot y}{\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot y, y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right)}, y, 1\right) \]

   if 1 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. sub-neg N/A

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

      11. unpow2 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 75.0

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

   5. Applied rewrites 75.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(0.008333333333333333 \cdot x, x, -0.16666666666666666\right), x\right)} \cdot \frac{\sinh y}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \cdot \sin x \leq -\infty:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot -0.16666666666666666\right) \cdot \frac{\sinh y}{y}\\ \mathbf{elif}\;\frac{\sinh y}{y} \cdot \sin x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot \left(y \cdot y\right), \mathsf{fma}\left(0.0001984126984126984 \cdot y, y, 0.008333333333333333\right) \cdot \mathsf{fma}\left(0.0001984126984126984 \cdot y, y, 0.008333333333333333\right), -0.027777777777777776\right) \cdot y}{\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984 \cdot y, y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right)}, y, 1\right) \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(0.008333333333333333 \cdot x, x, -0.16666666666666666\right), x\right) \cdot \frac{\sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Reproduce

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