Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.3% → 99.8%

Time: 5.0s

Alternatives: 20

Speedup: 1.4×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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)) / x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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)) / x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 = (sinh(y) / x) * sin(x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sinh.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-sinh.f64 N/A

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

   10. lift-sin.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 50.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq -1 \cdot 10^{-272} \lor \neg \left(t\_0 \leq 5 \cdot 10^{-309} \lor \neg \left(t\_0 \leq 2 \cdot 10^{-7}\right)\right)\right):\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x)))
   (if (or (<= t_0 (- INFINITY))
           (not
            (or (<= t_0 -1e-272)
                (not (or (<= t_0 5e-309) (not (<= t_0 2e-7)))))))
     (* (* (* y y) 0.16666666666666666) y)
     y)))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !((t_0 <= -1e-272) || !((t_0 <= 5e-309) || !(t_0 <= 2e-7)))) {
		tmp = ((y * y) * 0.16666666666666666) * y;
	} else {
		tmp = y;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (Math.sin(x) * Math.sinh(y)) / x;
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !((t_0 <= -1e-272) || !((t_0 <= 5e-309) || !(t_0 <= 2e-7)))) {
		tmp = ((y * y) * 0.16666666666666666) * y;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sin(x) * math.sinh(y)) / x
	tmp = 0
	if (t_0 <= -math.inf) or not ((t_0 <= -1e-272) or not ((t_0 <= 5e-309) or not (t_0 <= 2e-7))):
		tmp = ((y * y) * 0.16666666666666666) * y
	else:
		tmp = y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !((t_0 <= -1e-272) || !((t_0 <= 5e-309) || !(t_0 <= 2e-7))))
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y);
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sin(x) * sinh(y)) / x;
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~(((t_0 <= -1e-272) || ~(((t_0 <= 5e-309) || ~((t_0 <= 2e-7)))))))
		tmp = ((y * y) * 0.16666666666666666) * y;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[Or[LessEqual[t$95$0, -1e-272], N[Not[Or[LessEqual[t$95$0, 5e-309], N[Not[LessEqual[t$95$0, 2e-7]], $MachinePrecision]]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -inf.0 or -9.9999999999999993e-273 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 4.9999999999999995e-309 or 1.9999999999999999e-7 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 87.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 47.8

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

   8. Applied rewrites 47.8%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 52.0

          \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]

   11. Applied rewrites 52.0%

       \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]

   if -inf.0 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -9.9999999999999993e-273 or 4.9999999999999995e-309 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 1.9999999999999999e-7

   1. Initial program 97.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \frac{1}{2} \]

      4. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

      6. lift-sinh.f64 72.0

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

      \[\leadsto y \]
   7. Step-by-step derivation

   8. Applied rewrites 71.9%

      \[\leadsto y \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin x \cdot \sinh y}{x} \leq -\infty \lor \neg \left(\frac{\sin x \cdot \sinh y}{x} \leq -1 \cdot 10^{-272} \lor \neg \left(\frac{\sin x \cdot \sinh y}{x} \leq 5 \cdot 10^{-309} \lor \neg \left(\frac{\sin x \cdot \sinh y}{x} \leq 2 \cdot 10^{-7}\right)\right)\right):\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-7}:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.1e-7)
   (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5))
   (/
    (*
     (sin x)
     (*
      (fma
       (fma
        (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        0.16666666666666666)
       (* y y)
       1.0)
      y))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.1e-7) {
		tmp = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	} else {
		tmp = (sin(x) * (fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.1e-7)
		tmp = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	else
		tmp = Float64(Float64(sin(x) * Float64(fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.1e-7], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * 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]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001e-7

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 81.9

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   5. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]

   if 1.1000000000000001e-7 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 90.5

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

   5. Applied rewrites 90.5%

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

Alternative 4: 79.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-7}:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.0001984126984126984, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.1e-7)
   (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5))
   (/
    (*
     (sin x)
     (*
      (fma
       (fma (* (* y y) 0.0001984126984126984) (* y y) 0.16666666666666666)
       (* y y)
       1.0)
      y))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.1e-7) {
		tmp = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	} else {
		tmp = (sin(x) * (fma(fma(((y * y) * 0.0001984126984126984), (y * y), 0.16666666666666666), (y * y), 1.0) * y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.1e-7)
		tmp = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	else
		tmp = Float64(Float64(sin(x) * Float64(fma(fma(Float64(Float64(y * y) * 0.0001984126984126984), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.1e-7], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001e-7

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 81.9

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   5. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]

   if 1.1000000000000001e-7 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 90.5

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

   5. Applied rewrites 90.5%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 90.5

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

   8. Applied rewrites 90.5%

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

Alternative 5: 92.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ t_1 := \frac{\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \sin x}{x} \cdot y\\ \mathbf{if}\;y \leq -4.7 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -0.24:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 17000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot \sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5)))
        (t_1 (* (/ (* (* (* y y) 0.16666666666666666) (sin x)) x) y)))
   (if (<= y -4.7e+104)
     t_1
     (if (<= y -0.24)
       t_0
       (if (<= y 17000.0)
         (* (/ (* (fma y (* y 0.16666666666666666) 1.0) (sin x)) x) y)
         (if (<= y 4e+104) t_0 t_1))))))

C

double code(double x, double y) {
	double t_0 = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	double t_1 = ((((y * y) * 0.16666666666666666) * sin(x)) / x) * y;
	double tmp;
	if (y <= -4.7e+104) {
		tmp = t_1;
	} else if (y <= -0.24) {
		tmp = t_0;
	} else if (y <= 17000.0) {
		tmp = ((fma(y, (y * 0.16666666666666666), 1.0) * sin(x)) / x) * y;
	} else if (y <= 4e+104) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5))
	t_1 = Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * sin(x)) / x) * y)
	tmp = 0.0
	if (y <= -4.7e+104)
		tmp = t_1;
	elseif (y <= -0.24)
		tmp = t_0;
	elseif (y <= 17000.0)
		tmp = Float64(Float64(Float64(fma(y, Float64(y * 0.16666666666666666), 1.0) * sin(x)) / x) * y);
	elseif (y <= 4e+104)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -4.7e+104], t$95$1, If[LessEqual[y, -0.24], t$95$0, If[LessEqual[y, 17000.0], N[(N[(N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 4e+104], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.70000000000000017e104 or 4e104 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 95.4

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

   8. Applied rewrites 95.4%

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

   if -4.70000000000000017e104 < y < -0.23999999999999999 or 17000 < y < 4e104

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 86.4

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   5. Applied rewrites 86.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]

   if -0.23999999999999999 < y < 17000

   1. Initial program 79.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 98.5

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

   7. Applied rewrites 98.5%

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

Alternative 6: 92.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ t_1 := \frac{\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \sin x}{x} \cdot y\\ \mathbf{if}\;y \leq -4.7 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -0.15:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 17000:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5)))
        (t_1 (* (/ (* (* (* y y) 0.16666666666666666) (sin x)) x) y)))
   (if (<= y -4.7e+104)
     t_1
     (if (<= y -0.15)
       t_0
       (if (<= y 17000.0) (* (/ (sin x) x) y) (if (<= y 4e+104) t_0 t_1))))))

C

double code(double x, double y) {
	double t_0 = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	double t_1 = ((((y * y) * 0.16666666666666666) * sin(x)) / x) * y;
	double tmp;
	if (y <= -4.7e+104) {
		tmp = t_1;
	} else if (y <= -0.15) {
		tmp = t_0;
	} else if (y <= 17000.0) {
		tmp = (sin(x) / x) * y;
	} else if (y <= 4e+104) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5))
	t_1 = Float64(Float64(Float64(Float64(Float64(y * y) * 0.16666666666666666) * sin(x)) / x) * y)
	tmp = 0.0
	if (y <= -4.7e+104)
		tmp = t_1;
	elseif (y <= -0.15)
		tmp = t_0;
	elseif (y <= 17000.0)
		tmp = Float64(Float64(sin(x) / x) * y);
	elseif (y <= 4e+104)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -4.7e+104], t$95$1, If[LessEqual[y, -0.15], t$95$0, If[LessEqual[y, 17000.0], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 4e+104], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.70000000000000017e104 or 4e104 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 95.4

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

   8. Applied rewrites 95.4%

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

   if -4.70000000000000017e104 < y < -0.149999999999999994 or 17000 < y < 4e104

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 86.4

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   5. Applied rewrites 86.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]

   if -0.149999999999999994 < y < 17000

   1. Initial program 79.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 7: 78.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-7}:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.1e-7)
   (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5))
   (/
    (*
     (sin x)
     (*
      (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)
      y))
    x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.1e-7) {
		tmp = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	} else {
		tmp = (sin(x) * (fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) * y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.1e-7)
		tmp = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5));
	else
		tmp = Float64(Float64(sin(x) * Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) * y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.1e-7], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001e-7

   1. Initial program 85.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 81.9

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   5. Applied rewrites 81.9%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]

   if 1.1000000000000001e-7 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 87.8

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

   5. Applied rewrites 87.8%

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

Alternative 8: 86.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)\\ \mathbf{if}\;y \leq -0.15:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 17000:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{elif}\;y \leq 10^{+263}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 2.0 (sinh y)) (fma (* x x) -0.08333333333333333 0.5))))
   (if (<= y -0.15)
     t_0
     (if (<= y 17000.0)
       (* (/ (sin x) x) y)
       (if (<= y 1e+263) t_0 (* (* (* y y) 0.16666666666666666) y))))))

C

double code(double x, double y) {
	double t_0 = (2.0 * sinh(y)) * fma((x * x), -0.08333333333333333, 0.5);
	double tmp;
	if (y <= -0.15) {
		tmp = t_0;
	} else if (y <= 17000.0) {
		tmp = (sin(x) / x) * y;
	} else if (y <= 1e+263) {
		tmp = t_0;
	} else {
		tmp = ((y * y) * 0.16666666666666666) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(2.0 * sinh(y)) * fma(Float64(x * x), -0.08333333333333333, 0.5))
	tmp = 0.0
	if (y <= -0.15)
		tmp = t_0;
	elseif (y <= 17000.0)
		tmp = Float64(Float64(sin(x) / x) * y);
	elseif (y <= 1e+263)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.08333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.15], t$95$0, If[LessEqual[y, 17000.0], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1e+263], t$95$0, N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.149999999999999994 or 17000 < y < 1.00000000000000002e263

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({x}^{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)\right) + \frac{1}{2} \cdot \left(e^{y} - \frac{1}{e^{y}}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{-1}{12} \cdot {x}^{2}\right) \cdot \left(e^{y} - \frac{1}{e^{y}}\right) + \color{blue}{\frac{1}{2}} \cdot \left(e^{y} - \frac{1}{e^{y}}\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{1}{2}\right)} \]

      4. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      5. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{1}{2}\right) \]

      7. lift-sinh.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{1}{2}\right) \]

      8. *-commutative N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \left({x}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left({x}^{2}, \color{blue}{\frac{-1}{12}}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, \frac{-1}{12}, \frac{1}{2}\right) \]

      11. lower-*.f64 84.4

          \[\leadsto \left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right) \]

   5. Applied rewrites 84.4%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot \mathsf{fma}\left(x \cdot x, -0.08333333333333333, 0.5\right)} \]

   if -0.149999999999999994 < y < 17000

   1. Initial program 79.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if 1.00000000000000002e263 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 100.0

          \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]

   11. Applied rewrites 100.0%

       \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 84.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -800:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot -0.0001984126984126984\right) \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right)}{x}\\ \mathbf{elif}\;y \leq 10^{-7}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sinh y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -800.0)
   (/
    (*
     (*
      (fma
       (- (* (* (* x x) -0.0001984126984126984) (* x x)) 0.16666666666666666)
       (* x x)
       1.0)
      x)
     (*
      (fma
       (fma
        (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        0.16666666666666666)
       (* y y)
       1.0)
      y))
    x)
   (if (<= y 1e-7) (* (/ (sin x) x) y) (* (* 2.0 (sinh y)) 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -800.0) {
		tmp = ((fma(((((x * x) * -0.0001984126984126984) * (x * x)) - 0.16666666666666666), (x * x), 1.0) * x) * (fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * y)) / x;
	} else if (y <= 1e-7) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = (2.0 * sinh(y)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -800.0)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(Float64(x * x) * -0.0001984126984126984) * Float64(x * x)) - 0.16666666666666666), Float64(x * x), 1.0) * x) * Float64(fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y)) / x);
	elseif (y <= 1e-7)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = Float64(Float64(2.0 * sinh(y)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -800.0], N[(N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * 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]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1e-7], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(N[(2.0 * N[Sinh[y], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -800

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 78.9%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 78.9

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

   11. Applied rewrites 78.9%

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

   if -800 < y < 9.9999999999999995e-8

   1. Initial program 79.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 9.9999999999999995e-8 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \frac{1}{2} \]

      4. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

      6. lift-sinh.f64 75.9

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 69.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-10}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot -0.0001984126984126984\right) \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.0001984126984126984 \cdot y, y, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot y, 1\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3e-10)
   (/
    (*
     (*
      (fma
       (- (* (* (* x x) -0.0001984126984126984) (* x x)) 0.16666666666666666)
       (* x x)
       1.0)
      x)
     (*
      (fma
       (fma
        (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        0.16666666666666666)
       (* y y)
       1.0)
      y))
    x)
   (*
    x
    (/
     (*
      (fma
       (fma
        (* y y)
        (fma (* 0.0001984126984126984 y) y 0.008333333333333333)
        0.16666666666666666)
       (* y y)
       1.0)
      y)
     x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3e-10) {
		tmp = ((fma(((((x * x) * -0.0001984126984126984) * (x * x)) - 0.16666666666666666), (x * x), 1.0) * x) * (fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * y)) / x;
	} else {
		tmp = x * ((fma(fma((y * y), fma((0.0001984126984126984 * y), y, 0.008333333333333333), 0.16666666666666666), (y * y), 1.0) * y) / x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3e-10)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(Float64(x * x) * -0.0001984126984126984) * Float64(x * x)) - 0.16666666666666666), Float64(x * x), 1.0) * x) * Float64(fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y)) / x);
	else
		tmp = Float64(x * Float64(Float64(fma(fma(Float64(y * y), fma(Float64(0.0001984126984126984 * y), y, 0.008333333333333333), 0.16666666666666666), Float64(y * y), 1.0) * y) / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3e-10], N[(N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * 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]), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(0.0001984126984126984 * y), $MachinePrecision] * y + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 78.9%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 78.9

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

   11. Applied rewrites 78.9%

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

   if -3e-10 < y

   1. Initial program 86.1%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 81.1

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 77.4

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

   10. Applied rewrites 77.4%

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

Alternative 11: 69.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+20}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.0001984126984126984 \cdot y, y, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot y, 1\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.02e+20)
   (/
    (*
     (* (fma -0.16666666666666666 (* x x) 1.0) x)
     (*
      (fma
       (fma
        (fma 0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        0.16666666666666666)
       (* y y)
       1.0)
      y))
    x)
   (*
    x
    (/
     (*
      (fma
       (fma
        (* y y)
        (fma (* 0.0001984126984126984 y) y 0.008333333333333333)
        0.16666666666666666)
       (* y y)
       1.0)
      y)
     x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.02e+20) {
		tmp = ((fma(-0.16666666666666666, (x * x), 1.0) * x) * (fma(fma(fma(0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * y)) / x;
	} else {
		tmp = x * ((fma(fma((y * y), fma((0.0001984126984126984 * y), y, 0.008333333333333333), 0.16666666666666666), (y * y), 1.0) * y) / x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.02e+20)
		tmp = Float64(Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x) * Float64(fma(fma(fma(0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y)) / x);
	else
		tmp = Float64(x * Float64(Float64(fma(fma(Float64(y * y), fma(Float64(0.0001984126984126984 * y), y, 0.008333333333333333), 0.16666666666666666), Float64(y * y), 1.0) * y) / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.02e+20], N[(N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * 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]), $MachinePrecision] / x), $MachinePrecision], N[(x * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(0.0001984126984126984 * y), $MachinePrecision] * y + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.02e20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 79.4

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

   8. Applied rewrites 79.4%

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

   if -1.02e20 < y

   1. Initial program 86.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 79.9

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 46.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 76.7

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

   10. Applied rewrites 76.7%

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

Alternative 12: 69.8% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+141}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right)}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(0.0001984126984126984 \cdot y, y, 0.008333333333333333\right), 0.16666666666666666\right), y \cdot y, 1\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.7e+141)
   (*
    (/
     (*
      (fma (* y y) 0.16666666666666666 1.0)
      (* (fma -0.16666666666666666 (* x x) 1.0) x))
     x)
    y)
   (*
    x
    (/
     (*
      (fma
       (fma
        (* y y)
        (fma (* 0.0001984126984126984 y) y 0.008333333333333333)
        0.16666666666666666)
       (* y y)
       1.0)
      y)
     x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.7e+141) {
		tmp = ((fma((y * y), 0.16666666666666666, 1.0) * (fma(-0.16666666666666666, (x * x), 1.0) * x)) / x) * y;
	} else {
		tmp = x * ((fma(fma((y * y), fma((0.0001984126984126984 * y), y, 0.008333333333333333), 0.16666666666666666), (y * y), 1.0) * y) / x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.7e+141)
		tmp = Float64(Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x)) / x) * y);
	else
		tmp = Float64(x * Float64(Float64(fma(fma(Float64(y * y), fma(Float64(0.0001984126984126984 * y), y, 0.008333333333333333), 0.16666666666666666), Float64(y * y), 1.0) * y) / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.7e+141], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(x * N[(N[(N[(N[(N[(y * y), $MachinePrecision] * N[(N[(0.0001984126984126984 * y), $MachinePrecision] * y + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.69999999999999979e141

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 88.9

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

   8. Applied rewrites 88.9%

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

   if -4.69999999999999979e141 < y

   1. Initial program 87.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 80.1

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 75.0

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

   10. Applied rewrites 75.0%

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

Alternative 13: 62.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.22 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+78}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, -0.0001984126984126984, 0.008333333333333333\right) \cdot x\right) \cdot x - 0.16666666666666666, x \cdot x, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.22e+45)
   (*
    (fma
     (fma
      (fma (* y y) 0.0001984126984126984 0.008333333333333333)
      (* y y)
      0.16666666666666666)
     (* y y)
     1.0)
    y)
   (if (<= x 2.35e+78)
     (*
      (fma
       (-
        (* (* (fma (* x x) -0.0001984126984126984 0.008333333333333333) x) x)
        0.16666666666666666)
       (* x x)
       1.0)
      y)
     (* (* (* y y) 0.16666666666666666) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.22e+45) {
		tmp = fma(fma(fma((y * y), 0.0001984126984126984, 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * y;
	} else if (x <= 2.35e+78) {
		tmp = fma((((fma((x * x), -0.0001984126984126984, 0.008333333333333333) * x) * x) - 0.16666666666666666), (x * x), 1.0) * y;
	} else {
		tmp = ((y * y) * 0.16666666666666666) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.22e+45)
		tmp = Float64(fma(fma(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y);
	elseif (x <= 2.35e+78)
		tmp = Float64(fma(Float64(Float64(Float64(fma(Float64(x * x), -0.0001984126984126984, 0.008333333333333333) * x) * x) - 0.16666666666666666), Float64(x * x), 1.0) * y);
	else
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.22e+45], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 2.35e+78], N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.21999999999999997e45

   1. Initial program 86.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \frac{1}{2} \]

      4. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

      6. lift-sinh.f64 76.7

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

   8. Applied rewrites 72.8%

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

   if 1.21999999999999997e45 < x < 2.35000000000000003e78

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

   8. Applied rewrites 34.3%

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

   if 2.35000000000000003e78 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 19.0

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

   8. Applied rewrites 19.0%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 50.0

          \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]

   11. Applied rewrites 50.0%

       \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 61.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.3 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.0001984126984126984, 0.008333333333333333\right), y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+78}:\\ \;\;\;\;\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.3e+42)
   (*
    (fma
     (fma
      (fma (* y y) 0.0001984126984126984 0.008333333333333333)
      (* y y)
      0.16666666666666666)
     (* y y)
     1.0)
    y)
   (if (<= x 2.35e+78)
     (* (* (fma (* x x) -0.16666666666666666 1.0) x) (/ y x))
     (* (* (* y y) 0.16666666666666666) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.3e+42) {
		tmp = fma(fma(fma((y * y), 0.0001984126984126984, 0.008333333333333333), (y * y), 0.16666666666666666), (y * y), 1.0) * y;
	} else if (x <= 2.35e+78) {
		tmp = (fma((x * x), -0.16666666666666666, 1.0) * x) * (y / x);
	} else {
		tmp = ((y * y) * 0.16666666666666666) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.3e+42)
		tmp = Float64(fma(fma(fma(Float64(y * y), 0.0001984126984126984, 0.008333333333333333), Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * y);
	elseif (x <= 2.35e+78)
		tmp = Float64(Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x) * Float64(y / x));
	else
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.3e+42], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 2.35e+78], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.2999999999999998e42

   1. Initial program 86.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \frac{1}{2} \]

      4. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

      6. lift-sinh.f64 77.1

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 77.1%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

   8. Applied rewrites 73.1%

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

   if 4.2999999999999998e42 < x < 2.35000000000000003e78

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 31.5

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

   8. Applied rewrites 31.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right) \cdot x\right) \cdot \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{y}{x}\right)\right) \]

   10. Applied rewrites 59.2%

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

   if 2.35000000000000003e78 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 19.0

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

   8. Applied rewrites 19.0%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 50.0

          \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]

   11. Applied rewrites 50.0%

       \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 60.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.3 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+78}:\\ \;\;\;\;\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\right) \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.3e+42)
   (*
    (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)
    y)
   (if (<= x 2.35e+78)
     (* (* (fma (* x x) -0.16666666666666666 1.0) x) (/ y x))
     (* (* (* y y) 0.16666666666666666) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.3e+42) {
		tmp = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) * y;
	} else if (x <= 2.35e+78) {
		tmp = (fma((x * x), -0.16666666666666666, 1.0) * x) * (y / x);
	} else {
		tmp = ((y * y) * 0.16666666666666666) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.3e+42)
		tmp = Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) * y);
	elseif (x <= 2.35e+78)
		tmp = Float64(Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x) * Float64(y / x));
	else
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.3e+42], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 2.35e+78], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.2999999999999998e42

   1. Initial program 86.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \frac{1}{2} \]

      4. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

      6. lift-sinh.f64 77.1

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 77.1%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 71.6

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

   8. Applied rewrites 71.6%

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

   if 4.2999999999999998e42 < x < 2.35000000000000003e78

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 31.5

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

   8. Applied rewrites 31.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x \cdot x, \frac{-1}{6}, 1\right) \cdot x\right) \cdot \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{y}{x}\right)\right) \]

   10. Applied rewrites 59.2%

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

   if 2.35000000000000003e78 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 19.0

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

   8. Applied rewrites 19.0%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 50.0

          \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]

   11. Applied rewrites 50.0%

       \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 60.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.22 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+78}:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot x\right) \cdot y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.22e+45)
   (*
    (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)
    y)
   (if (<= x 1.95e+78)
     (/ (* (* (* (* x x) -0.16666666666666666) x) y) x)
     (* (* (* y y) 0.16666666666666666) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.22e+45) {
		tmp = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) * y;
	} else if (x <= 1.95e+78) {
		tmp = ((((x * x) * -0.16666666666666666) * x) * y) / x;
	} else {
		tmp = ((y * y) * 0.16666666666666666) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.22e+45)
		tmp = Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) * y);
	elseif (x <= 1.95e+78)
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * -0.16666666666666666) * x) * y) / x);
	else
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.22e+45], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 1.95e+78], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.21999999999999997e45

   1. Initial program 86.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \frac{1}{2} \]

      4. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

      6. lift-sinh.f64 76.7

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 71.3

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

   8. Applied rewrites 71.3%

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

   if 1.21999999999999997e45 < x < 1.9500000000000002e78

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 35.8

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

   8. Applied rewrites 35.8%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 35.8

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

   11. Applied rewrites 35.8%

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

   if 1.9500000000000002e78 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 19.0

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

   8. Applied rewrites 19.0%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 50.0

          \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]

   11. Applied rewrites 50.0%

       \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 60.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.6e+42)
   (*
    (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)
    y)
   (* (* (* y y) 0.16666666666666666) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.6e+42) {
		tmp = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) * y;
	} else {
		tmp = ((y * y) * 0.16666666666666666) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.6e+42)
		tmp = Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) * y);
	else
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.6e+42], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5999999999999999e42

   1. Initial program 86.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

      3. rec-exp N/A

         \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \frac{1}{2} \]

      4. sinh-undef N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

      6. lift-sinh.f64 77.1

         \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

   5. Applied rewrites 77.1%

      \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 71.6

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

   8. Applied rewrites 71.6%

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

   if 2.5999999999999999e42 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 17.5

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

   8. Applied rewrites 17.5%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 49.3

          \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]

   11. Applied rewrites 49.3%

       \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 52.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+99} \lor \neg \left(y \leq 1.7 \cdot 10^{+91}\right):\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, -0.16666666666666666, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.6e+99) (not (<= y 1.7e+91)))
   (* (* (* y y) 0.16666666666666666) y)
   (fma (* (* x x) y) -0.16666666666666666 y)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.6e+99) || !(y <= 1.7e+91)) {
		tmp = ((y * y) * 0.16666666666666666) * y;
	} else {
		tmp = fma(((x * x) * y), -0.16666666666666666, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.6e+99) || !(y <= 1.7e+91))
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y);
	else
		tmp = fma(Float64(Float64(x * x) * y), -0.16666666666666666, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.6e+99], N[Not[LessEqual[y, 1.7e+91]], $MachinePrecision]], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * y), $MachinePrecision] * -0.16666666666666666 + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.6000000000000002e99 or 1.7e91 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 72.4

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

   8. Applied rewrites 72.4%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 72.4

          \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]

   11. Applied rewrites 72.4%

       \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]

   if -3.6000000000000002e99 < y < 1.7e91

   1. Initial program 84.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sin.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 50.0

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

   8. Applied rewrites 50.0%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+99} \lor \neg \left(y \leq 1.7 \cdot 10^{+91}\right):\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot y, -0.16666666666666666, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 56.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(y, y \cdot 0.16666666666666666, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.6e+41)
   (* (fma y (* y 0.16666666666666666) 1.0) y)
   (* (* (* y y) 0.16666666666666666) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.6e+41) {
		tmp = fma(y, (y * 0.16666666666666666), 1.0) * y;
	} else {
		tmp = ((y * y) * 0.16666666666666666) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.6e+41)
		tmp = Float64(fma(y, Float64(y * 0.16666666666666666), 1.0) * y);
	else
		tmp = Float64(Float64(Float64(y * y) * 0.16666666666666666) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.6e+41], N[(N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.6000000000000001e41

   1. Initial program 86.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 65.8

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

   8. Applied rewrites 65.8%

      \[\leadsto \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 65.8

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

   10. Applied rewrites 65.8%

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

   if 2.6000000000000001e41 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 17.5

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

   8. Applied rewrites 17.5%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 49.3

          \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]

   11. Applied rewrites 49.3%

       \[\leadsto \left(\left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 27.1% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 y)

C

double code(double x, double y) {
	return 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 = y
end function

Java

public static double code(double x, double y) {
	return y;
}

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

function tmp = code(x, y)
	tmp = y;
end

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 89.6%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(e^{y} - \frac{1}{e^{y}}\right) \cdot \color{blue}{\frac{1}{2}} \]

   3. rec-exp N/A

      \[\leadsto \left(e^{y} - e^{\mathsf{neg}\left(y\right)}\right) \cdot \frac{1}{2} \]

   4. sinh-undef N/A

      \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(2 \cdot \sinh y\right) \cdot \frac{1}{2} \]

   6. lift-sinh.f64 63.7

      \[\leadsto \left(2 \cdot \sinh y\right) \cdot 0.5 \]

5. Applied rewrites 63.7%

   \[\leadsto \color{blue}{\left(2 \cdot \sinh y\right) \cdot 0.5} \]

6. Taylor expanded in y around 0

   \[\leadsto y \]
7. Step-by-step derivation

8. Applied rewrites 31.4%

   \[\leadsto y \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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) / x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025072 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

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

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