Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.4% → 99.9%

Time: 11.1s

Alternatives: 23

Speedup: 1.0×

• 50.1% 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: 89.4% 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: 99.9% 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 89.9%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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: 85.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y \cdot \sin x}{x}\\ t_1 := \frac{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.005:\\ \;\;\;\;\sin x \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sinh y) (sin x)) x))
        (t_1
         (/
          (fma
           (* y y)
           (* y (fma (* y y) 0.008333333333333333 0.16666666666666666))
           y)
          x)))
   (if (<= t_0 (- INFINITY))
     (*
      t_1
      (fma
       (fma
        (* x x)
        (fma x (* x -0.0001984126984126984) 0.008333333333333333)
        -0.16666666666666666)
       (* x (* x x))
       x))
     (if (<= t_0 0.005) (* (sin x) t_1) (/ (sinh y) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (sinh(y) * sin(x)) / x;
	double t_1 = fma((y * y), (y * fma((y * y), 0.008333333333333333, 0.16666666666666666)), y) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1 * fma(fma((x * x), fma(x, (x * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666), (x * (x * x)), x);
	} else if (t_0 <= 0.005) {
		tmp = sin(x) * t_1;
	} else {
		tmp = sinh(y) / 1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sinh(y) * sin(x)) / x)
	t_1 = Float64(fma(Float64(y * y), Float64(y * fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666)), y) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(t_1 * fma(fma(Float64(x * x), fma(x, Float64(x * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666), Float64(x * Float64(x * x)), x));
	elseif (t_0 <= 0.005)
		tmp = Float64(sin(x) * t_1);
	else
		tmp = Float64(sinh(y) / 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sinh[y], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y * y), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(t$95$1 * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.005], N[(N[Sin[x], $MachinePrecision] * t$95$1), $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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 85.5

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. unpow2 N/A

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

      7. unpow3 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right) \cdot y, y\right)}{x} \cdot \left(\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{x}^{3}} + x\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{1}{6}\right) \cdot y, y\right)}{x} \cdot \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}, {x}^{3}, x\right)} \]

   10. Applied rewrites 70.6%

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

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 0.0050000000000000001

   1. Initial program 78.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 98.9

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

   7. Applied rewrites 98.9%

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

   if 0.0050000000000000001 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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. 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 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y \cdot \sin x}{x} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{elif}\;\frac{\sinh y \cdot \sin x}{x} \leq 0.005:\\ \;\;\;\;\sin x \cdot \frac{\mathsf{fma}\left(y \cdot y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{1}\\ \end{array} \]
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 2024228 
(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))