Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 99.0%

Time: 3.2s

Alternatives: 14

Speedup: 1.0×

• 49.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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: 99.0% accurate, 0.4× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (let* ((t_0
         (fma
          (fma (* y y) 0.008333333333333333 0.16666666666666666)
          (* y y)
          1.0))
        (t_1 (/ (sinh y) y))
        (t_2 (* (sin x_m) t_1)))
   (*
    x_s
    (if (<= t_2 (- INFINITY))
      (* (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m) t_0)
      (if (<= t_2 1.0) (* (sin x_m) t_0) (* x_m t_1))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double t_0 = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0);
	double t_1 = sinh(y) / y;
	double t_2 = sin(x_m) * t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m) * t_0;
	} else if (t_2 <= 1.0) {
		tmp = sin(x_m) * t_0;
	} else {
		tmp = x_m * t_1;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	t_0 = fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)
	t_1 = Float64(sinh(y) / y)
	t_2 = Float64(sin(x_m) * t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m) * t_0);
	elseif (t_2 <= 1.0)
		tmp = Float64(sin(x_m) * t_0);
	else
		tmp = Float64(x_m * t_1);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := Block[{t$95$0 = N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[x$95$m], $MachinePrecision] * t$95$1), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$2, 1.0], N[(N[Sin[x$95$m], $MachinePrecision] * t$95$0), $MachinePrecision], N[(x$95$m * t$95$1), $MachinePrecision]]]), $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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

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

       7. lift-fma.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-fma.f64 N/A

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

       10. lift-*.f64 96.5

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

   11. Applied rewrites 96.5%

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

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

   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

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

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.9% accurate, 0.4× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (sin x_m) t_0)))
   (*
    x_s
    (if (<= t_1 (- INFINITY))
      (*
       (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m)
       (fma
        (fma (* y y) 0.008333333333333333 0.16666666666666666)
        (* y y)
        1.0))
      (if (<= t_1 1.0)
        (* (sin x_m) (fma (* y y) 0.16666666666666666 1.0))
        (* x_m t_0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = sin(x_m) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0);
	} else if (t_1 <= 1.0) {
		tmp = sin(x_m) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = x_m * t_0;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(sin(x_m) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0));
	elseif (t_1 <= 1.0)
		tmp = Float64(sin(x_m) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(x_m * t_0);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x$95$m], $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(N[Sin[x$95$m], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(x$95$m * t$95$0), $MachinePrecision]]]), $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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

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

       7. lift-fma.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-fma.f64 N/A

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

       10. lift-*.f64 96.5

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

   11. Applied rewrites 96.5%

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

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.0

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

   5. Applied rewrites 99.0%

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

   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

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

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.7% accurate, 0.4× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)) (t_1 (* (sin x_m) t_0)))
   (*
    x_s
    (if (<= t_1 (- INFINITY))
      (*
       (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m)
       (fma
        (fma (* y y) 0.008333333333333333 0.16666666666666666)
        (* y y)
        1.0))
      (if (<= t_1 1.0) (sin x_m) (* x_m t_0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double t_0 = sinh(y) / y;
	double t_1 = sin(x_m) * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0);
	} else if (t_1 <= 1.0) {
		tmp = sin(x_m);
	} else {
		tmp = x_m * t_0;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	t_0 = Float64(sinh(y) / y)
	t_1 = Float64(sin(x_m) * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0));
	elseif (t_1 <= 1.0)
		tmp = sin(x_m);
	else
		tmp = Float64(x_m * t_0);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x$95$m], $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[Sin[x$95$m], $MachinePrecision], N[(x$95$m * t$95$0), $MachinePrecision]]]), $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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

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

       7. lift-fma.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-fma.f64 N/A

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

       10. lift-*.f64 96.5

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

   11. Applied rewrites 96.5%

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

   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

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

      1. lift-sin.f64 98.5

         \[\leadsto \sin x \]

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\sin x} \]

   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

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

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 94.6% accurate, 0.4× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (let* ((t_0 (* (sin x_m) (/ (sinh y) y))))
   (*
    x_s
    (if (<= t_0 (- INFINITY))
      (*
       (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m)
       (fma
        (fma (* y y) 0.008333333333333333 0.16666666666666666)
        (* y y)
        1.0))
      (if (<= t_0 1.0)
        (sin x_m)
        (*
         x_m
         (/
          (*
           (fma
            (fma
             (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
             (* y y)
             0.16666666666666666)
            (* y y)
            1.0)
           y)
          y)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double t_0 = sin(x_m) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0);
	} else if (t_0 <= 1.0) {
		tmp = sin(x_m);
	} else {
		tmp = x_m * ((fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * y) / y);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	t_0 = Float64(sin(x_m) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0));
	elseif (t_0 <= 1.0)
		tmp = sin(x_m);
	else
		tmp = Float64(x_m * Float64(Float64(fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y) / y));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := Block[{t$95$0 = N[(N[Sin[x$95$m], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, (-Infinity)], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[x$95$m], $MachinePrecision], N[(x$95$m * N[(N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]), $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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

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

       7. lift-fma.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-fma.f64 N/A

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

       10. lift-*.f64 96.5

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

   11. Applied rewrites 96.5%

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

   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

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

      1. lift-sin.f64 98.5

         \[\leadsto \sin x \]

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\sin x} \]

   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

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 88.8

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

   8. Applied rewrites 88.8%

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y}}{y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 71.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m \cdot \frac{\sinh y}{y} \leq 0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot x\_m\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (* (sin x_m) (/ (sinh y) y)) 0.02)
    (*
     (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m)
     (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0))
    (*
     x_m
     (/
      (*
       (fma
        (fma
         (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
         (* y y)
         0.16666666666666666)
        (* y y)
        1.0)
       y)
      y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if ((sin(x_m) * (sinh(y) / y)) <= 0.02) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = x_m * ((fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * y) / y);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (Float64(sin(x_m) * Float64(sinh(y) / y)) <= 0.02)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(x_m * Float64(Float64(fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y) / y));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[(N[Sin[x$95$m], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.0200000000000000004

   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.1%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

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

       7. lift-fma.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-fma.f64 N/A

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

       10. lift-*.f64 74.6

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

   11. Applied rewrites 74.6%

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

   if 0.0200000000000000004 < (*.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

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 68.2

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

   8. Applied rewrites 68.2%

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y}}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 70.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m \cdot \frac{\sinh y}{y} \leq 0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot x\_m\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (* (sin x_m) (/ (sinh y) y)) 0.02)
    (*
     (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m)
     (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0))
    (*
     x_m
     (fma
      (fma
       (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
       (* y y)
       0.16666666666666666)
      (* y y)
      1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if ((sin(x_m) * (sinh(y) / y)) <= 0.02) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0);
	} else {
		tmp = x_m * fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (Float64(sin(x_m) * Float64(sinh(y) / y)) <= 0.02)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0));
	else
		tmp = Float64(x_m * fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[(N[Sin[x$95$m], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.0200000000000000004

   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.1%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

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

       7. lift-fma.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-fma.f64 N/A

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

       10. lift-*.f64 74.6

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

   11. Applied rewrites 74.6%

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

   if 0.0200000000000000004 < (*.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

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 3.9%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 66.9

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

   11. Applied rewrites 66.9%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m \cdot \frac{\sinh y}{y} \leq 0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot x\_m\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (* (sin x_m) (/ (sinh y) y)) 0.02)
    (*
     (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m)
     (fma (* y y) 0.16666666666666666 1.0))
    (*
     x_m
     (fma
      (fma
       (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
       (* y y)
       0.16666666666666666)
      (* y y)
      1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if ((sin(x_m) * (sinh(y) / y)) <= 0.02) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = x_m * fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (Float64(sin(x_m) * Float64(sinh(y) / y)) <= 0.02)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(x_m * fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[(N[Sin[x$95$m], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(N[(0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.0200000000000000004

   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.1%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. +-commutative N/A

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

       4. lift-fma.f64 N/A

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

       5. lift-*.f64 73.5

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

   11. Applied rewrites 73.5%

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

   if 0.0200000000000000004 < (*.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

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 3.9%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 N/A

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

       9. pow2 N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 66.9

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

   11. Applied rewrites 66.9%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m \cdot \frac{\sinh y}{y} \leq 0.02:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot x\_m\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (* (sin x_m) (/ (sinh y) y)) 0.02)
    (*
     (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m)
     (fma (* y y) 0.16666666666666666 1.0))
    (*
     x_m
     (fma
      (fma (* y y) 0.008333333333333333 0.16666666666666666)
      (* y y)
      1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if ((sin(x_m) * (sinh(y) / y)) <= 0.02) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = x_m * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (Float64(sin(x_m) * Float64(sinh(y) / y)) <= 0.02)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(x_m * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[(N[Sin[x$95$m], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.02], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 0.0200000000000000004

   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 75.9

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.1%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. +-commutative N/A

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

       4. lift-fma.f64 N/A

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

       5. lift-*.f64 73.5

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

   11. Applied rewrites 73.5%

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

   if 0.0200000000000000004 < (*.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

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 3.9%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

          \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) \cdot \left(y \cdot y\right) + 1\right) \]

       7. lift-fma.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-fma.f64 N/A

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

       10. lift-*.f64 62.3

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

   11. Applied rewrites 62.3%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\sin x\_m \cdot \frac{\sinh y}{y}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y) :precision binary64 (* x_s (* (sin x_m) (/ (sinh y) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	return x_s * (sin(x_m) * (sinh(y) / y));
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_s, x_m, y)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    code = x_s * (sin(x_m) * (sinh(y) / y))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y) {
	return x_s * (Math.sin(x_m) * (Math.sinh(y) / y));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y):
	return x_s * (math.sin(x_m) * (math.sinh(y) / y))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	return Float64(x_s * Float64(sin(x_m) * Float64(sinh(y) / y)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y)
	tmp = x_s * (sin(x_m) * (sinh(y) / y));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * N[(N[Sin[x$95$m], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

Alternative 10: 66.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m \leq -0.01:\\ \;\;\;\;\frac{\left(\left(\left(x\_m \cdot x\_m\right) \cdot -0.16666666666666666\right) \cdot x\_m\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (sin x_m) -0.01)
    (/ (* (* (* (* x_m x_m) -0.16666666666666666) x_m) y) y)
    (*
     x_m
     (fma
      (fma (* y y) 0.008333333333333333 0.16666666666666666)
      (* y y)
      1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if (sin(x_m) <= -0.01) {
		tmp = ((((x_m * x_m) * -0.16666666666666666) * x_m) * y) / y;
	} else {
		tmp = x_m * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (sin(x_m) <= -0.01)
		tmp = Float64(Float64(Float64(Float64(Float64(x_m * x_m) * -0.16666666666666666) * x_m) * y) / y);
	else
		tmp = Float64(x_m * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[Sin[x$95$m], $MachinePrecision], -0.01], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], N[(x$95$m * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 35.7%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 35.7

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

   11. Applied rewrites 35.7%

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot x\right) \cdot \frac{y}{y} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 43.8

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

   13. Applied rewrites 43.8%

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

   if -0.0100000000000000002 < (sin.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 35.9%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

          \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) \cdot \left(y \cdot y\right) + 1\right) \]

       7. lift-fma.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-fma.f64 N/A

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

       10. lift-*.f64 74.4

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

   11. Applied rewrites 74.4%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 64.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot x\_m\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (sin x_m) -0.01)
    (* (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m) 1.0)
    (*
     x_m
     (fma
      (fma (* y y) 0.008333333333333333 0.16666666666666666)
      (* y y)
      1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if (sin(x_m) <= -0.01) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m) * 1.0;
	} else {
		tmp = x_m * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (sin(x_m) <= -0.01)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m) * 1.0);
	else
		tmp = Float64(x_m * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[Sin[x$95$m], $MachinePrecision], -0.01], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * 1.0), $MachinePrecision], N[(x$95$m * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 35.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 35.7%

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

   if -0.0100000000000000002 < (sin.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 35.9%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. pow2 N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. pow2 N/A

          \[\leadsto x \cdot \left(\left(\left(y \cdot y\right) \cdot \frac{1}{120} + \frac{1}{6}\right) \cdot \left(y \cdot y\right) + 1\right) \]

       7. lift-fma.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-fma.f64 N/A

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

       10. lift-*.f64 74.4

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

   11. Applied rewrites 74.4%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 57.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\sin x\_m \leq -0.01:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x\_m \cdot x\_m, 1\right) \cdot x\_m\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (*
  x_s
  (if (<= (sin x_m) -0.01)
    (* (* (fma -0.16666666666666666 (* x_m x_m) 1.0) x_m) 1.0)
    (* x_m (fma (* y y) 0.16666666666666666 1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	double tmp;
	if (sin(x_m) <= -0.01) {
		tmp = (fma(-0.16666666666666666, (x_m * x_m), 1.0) * x_m) * 1.0;
	} else {
		tmp = x_m * fma((y * y), 0.16666666666666666, 1.0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	tmp = 0.0
	if (sin(x_m) <= -0.01)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x_m * x_m), 1.0) * x_m) * 1.0);
	else
		tmp = Float64(x_m * fma(Float64(y * y), 0.16666666666666666, 1.0));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * If[LessEqual[N[Sin[x$95$m], $MachinePrecision], -0.01], N[(N[(N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] * 1.0), $MachinePrecision], N[(x$95$m * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0100000000000000002

   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 35.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 35.7%

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

   if -0.0100000000000000002 < (sin.f64 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 35.9%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. pow2 N/A

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

       3. +-commutative N/A

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

       4. lift-fma.f64 N/A

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

       5. lift-*.f64 64.5

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

   11. Applied rewrites 64.5%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 48.5% accurate, 12.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y)
 :precision binary64
 (* x_s (* x_m (fma (* y y) 0.16666666666666666 1.0))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	return x_s * (x_m * fma((y * y), 0.16666666666666666, 1.0));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	return Float64(x_s * Float64(x_m * fma(Float64(y * y), 0.16666666666666666, 1.0)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * N[(x$95$m * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 63.1%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 27.1%

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

9. Taylor expanded in y around 0

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

    1. *-commutative N/A

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

    2. pow2 N/A

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

    3. +-commutative N/A

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

    4. lift-fma.f64 N/A

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

    5. lift-*.f64 48.5

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

11. Applied rewrites 48.5%

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

Alternative 14: 27.1% accurate, 36.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot 1\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y) :precision binary64 (* x_s (* x_m 1.0)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y) {
	return x_s * (x_m * 1.0);
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_s, x_m, y)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    code = x_s * (x_m * 1.0d0)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y) {
	return x_s * (x_m * 1.0);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y):
	return x_s * (x_m * 1.0)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y)
	return Float64(x_s * Float64(x_m * 1.0))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y)
	tmp = x_s * (x_m * 1.0);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_] := N[(x$95$s * N[(x$95$m * 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 63.1%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 27.1%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 27.1%

    \[\leadsto x \cdot \color{blue}{1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025089 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))