Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.9% → 99.8%

Time: 9.3s

Alternatives: 16

Speedup: 1.5×

• 49.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 88.9% 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 92.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 81.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin x \cdot \sinh y}{x}\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot t\_1\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t\_1 \cdot x\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (sin x) (sinh y)) x))
        (t_1
         (fma
          (fma (* y y) 0.008333333333333333 0.16666666666666666)
          (* y y)
          1.0)))
   (if (<= t_0 -2e-7)
     (*
      (*
       (fma
        (fma
         (fma -0.0001984126984126984 (* x x) 0.008333333333333333)
         (* x x)
         -0.16666666666666666)
        (* x x)
        1.0)
       t_1)
      y)
     (if (<= t_0 2e-50) (* (/ (sin x) x) y) (/ (* (* t_1 x) y) x)))))

C

double code(double x, double y) {
	double t_0 = (sin(x) * sinh(y)) / x;
	double t_1 = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0);
	double tmp;
	if (t_0 <= -2e-7) {
		tmp = (fma(fma(fma(-0.0001984126984126984, (x * x), 0.008333333333333333), (x * x), -0.16666666666666666), (x * x), 1.0) * t_1) * y;
	} else if (t_0 <= 2e-50) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = ((t_1 * x) * y) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(sin(x) * sinh(y)) / x)
	t_1 = fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)
	tmp = 0.0
	if (t_0 <= -2e-7)
		tmp = Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(x * x), 0.008333333333333333), Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * t_1) * y);
	elseif (t_0 <= 2e-50)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = Float64(Float64(Float64(t_1 * x) * y) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sin[x], $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-7], N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 2e-50], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(t$95$1 * x), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < -1.9999999999999999e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.5%

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

   if -1.9999999999999999e-7 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x) < 2.00000000000000002e-50

   1. Initial program 84.1%

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

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 2.00000000000000002e-50 < (/.f64 (*.f64 (sin.f64 x) (sinh.f64 y)) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.2%

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

Alternative 3: 93.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.225:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right)}{x} \cdot y\right) \cdot \sin x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\left(e^{y} - e^{-y}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, 0.16666666666666666\right), y \cdot y, 1\right) \cdot \left(\sin x \cdot y\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.225)
   (*
    (*
     (/
      (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)
      x)
     y)
    (sin x))
   (if (<= y 5e+59)
     (* (- (exp y) (exp (- y))) 0.5)
     (/
      (*
       (fma (fma 0.008333333333333333 (* y y) 0.16666666666666666) (* y y) 1.0)
       (* (sin x) y))
      x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.225) {
		tmp = ((fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) / x) * y) * sin(x);
	} else if (y <= 5e+59) {
		tmp = (exp(y) - exp(-y)) * 0.5;
	} else {
		tmp = (fma(fma(0.008333333333333333, (y * y), 0.16666666666666666), (y * y), 1.0) * (sin(x) * y)) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.225)
		tmp = Float64(Float64(Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) / x) * y) * sin(x));
	elseif (y <= 5e+59)
		tmp = Float64(Float64(exp(y) - exp(Float64(-y))) * 0.5);
	else
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(y * y), 0.16666666666666666), Float64(y * y), 1.0) * Float64(sin(x) * y)) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.225], N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+59], N[(N[(N[Exp[y], $MachinePrecision] - N[Exp[(-y)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.225000000000000006

   1. Initial program 88.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 95.4%

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

   if 0.225000000000000006 < y < 4.9999999999999997e59

   1. Initial program 99.9%

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

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. rec-exp N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 76.7

         \[\leadsto \left(e^{y} - e^{\color{blue}{-y}}\right) \cdot 0.5 \]

   5. Applied rewrites 76.7%

      \[\leadsto \color{blue}{\left(e^{y} - e^{-y}\right) \cdot 0.5} \]

   if 4.9999999999999997e59 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.3%

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

   7. Applied rewrites 98.3%

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

Alternative 4: 91.2% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (*
   (/
    (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)
    x)
   y)
  (sin x)))

C

double code(double x, double y) {
	return ((fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) / x) * y) * sin(x);
}

Julia

function code(x, y)
	return Float64(Float64(Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) / x) * y) * sin(x))
end

Wolfram

code[x_, y_] := N[(N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 90.9%

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

Alternative 5: 89.2% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (*
  (*
   (/ (sin x) x)
   (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0))
  y))

C

double code(double x, double y) {
	return ((sin(x) / x) * fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0)) * y;
}

Julia

function code(x, y)
	return Float64(Float64(Float64(sin(x) / x) * fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0)) * y)
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 92.2%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 88.4%

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

Alternative 6: 88.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\sin x}{x} \cdot \mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.008333333333333333, y \cdot y, 1\right)\right) \cdot y \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (* (/ (sin x) x) (fma (* (* y y) 0.008333333333333333) (* y y) 1.0)) y))

C

double code(double x, double y) {
	return ((sin(x) / x) * fma(((y * y) * 0.008333333333333333), (y * y), 1.0)) * y;
}

Julia

function code(x, y)
	return Float64(Float64(Float64(sin(x) / x) * fma(Float64(Float64(y * y) * 0.008333333333333333), Float64(y * y), 1.0)) * y)
end

Wolfram

code[x_, y_] := N[(N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]

Derivation

1. Initial program 92.2%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 88.4%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 87.9%

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

Alternative 7: 71.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left({y}^{3}, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot \sin x\right) \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.4e-11)
   (fma (pow y 3.0) (fma (* y y) 0.008333333333333333 0.16666666666666666) y)
   (/ (* (* (fma (* y y) 0.16666666666666666 1.0) (sin x)) y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.4e-11) {
		tmp = fma(pow(y, 3.0), fma((y * y), 0.008333333333333333, 0.16666666666666666), y);
	} else {
		tmp = ((fma((y * y), 0.16666666666666666, 1.0) * sin(x)) * y) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.4e-11)
		tmp = fma((y ^ 3.0), fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), y);
	else
		tmp = Float64(Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * sin(x)) * y) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.4e-11], N[(N[Power[y, 3.0], $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] + y), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4000000000000001e-11

   1. Initial program 89.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in x 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

   8. Applied rewrites 61.7%

      \[\leadsto \mathsf{fma}\left({y}^{3}, \color{blue}{\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right)}, y\right) \]

   if 2.4000000000000001e-11 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 58.8

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

   5. Applied rewrites 58.8%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 82.7%

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

Alternative 8: 60.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+119}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{y}{x} \cdot x\right) \cdot x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.75e+63)
   (*
    (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)
    y)
   (if (<= x 7.4e+119)
     (* (* (* (fma -0.027777777777777776 (* x x) 0.16666666666666666) y) y) y)
     (/ (* (* (/ y x) x) x) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.75e+63) {
		tmp = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) * y;
	} else if (x <= 7.4e+119) {
		tmp = ((fma(-0.027777777777777776, (x * x), 0.16666666666666666) * y) * y) * y;
	} else {
		tmp = (((y / x) * x) * x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.75e+63)
		tmp = Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) * y);
	elseif (x <= 7.4e+119)
		tmp = Float64(Float64(Float64(fma(-0.027777777777777776, Float64(x * x), 0.16666666666666666) * y) * y) * y);
	else
		tmp = Float64(Float64(Float64(Float64(y / x) * x) * x) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.75e+63], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 7.4e+119], N[(N[(N[(N[(-0.027777777777777776 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(y / x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.75000000000000015e63

   1. Initial program 90.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.9%

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

   if 1.75000000000000015e63 < x < 7.3999999999999999e119

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 2.2%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot {x}^{2} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 17.6%

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

   12. Taylor expanded in y around inf

       \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \cdot y \]
   13. Step-by-step derivation

   14. Applied rewrites 48.6%

       \[\leadsto \left(\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot y \]

   if 7.3999999999999999e119 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 58.3

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 13.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 13.9%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 60.1%

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

Alternative 9: 56.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+119}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, x \cdot x, -0.16666666666666666\right), x \cdot x, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.75e+63)
   (*
    (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)
    y)
   (if (<= x 7.4e+119)
     (* (* (* (fma -0.027777777777777776 (* x x) 0.16666666666666666) y) y) y)
     (*
      (fma (fma 0.008333333333333333 (* x x) -0.16666666666666666) (* x x) 1.0)
      y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.75e+63) {
		tmp = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) * y;
	} else if (x <= 7.4e+119) {
		tmp = ((fma(-0.027777777777777776, (x * x), 0.16666666666666666) * y) * y) * y;
	} else {
		tmp = fma(fma(0.008333333333333333, (x * x), -0.16666666666666666), (x * x), 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.75e+63)
		tmp = Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) * y);
	elseif (x <= 7.4e+119)
		tmp = Float64(Float64(Float64(fma(-0.027777777777777776, Float64(x * x), 0.16666666666666666) * y) * y) * y);
	else
		tmp = Float64(fma(fma(0.008333333333333333, Float64(x * x), -0.16666666666666666), Float64(x * x), 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.75e+63], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 7.4e+119], N[(N[(N[(N[(-0.027777777777777776 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.75000000000000015e63

   1. Initial program 90.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.9%

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

   if 1.75000000000000015e63 < x < 7.3999999999999999e119

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 2.2%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot {x}^{2} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 17.6%

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

   12. Taylor expanded in y around inf

       \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \cdot y \]
   13. Step-by-step derivation

   14. Applied rewrites 48.6%

       \[\leadsto \left(\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot y \]

   if 7.3999999999999999e119 < x

   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

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 58.3

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 29.1%

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

Alternative 10: 56.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, 0.16666666666666666\right), y \cdot y, 1\right) \cdot y\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+119}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \left(-x\right) \cdot x, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.75e+63)
   (*
    (fma (fma (* y y) 0.008333333333333333 0.16666666666666666) (* y y) 1.0)
    y)
   (if (<= x 7.4e+119)
     (* (* (* (fma -0.027777777777777776 (* x x) 0.16666666666666666) y) y) y)
     (* (fma -0.16666666666666666 (* (- x) x) 1.0) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.75e+63) {
		tmp = fma(fma((y * y), 0.008333333333333333, 0.16666666666666666), (y * y), 1.0) * y;
	} else if (x <= 7.4e+119) {
		tmp = ((fma(-0.027777777777777776, (x * x), 0.16666666666666666) * y) * y) * y;
	} else {
		tmp = fma(-0.16666666666666666, (-x * x), 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.75e+63)
		tmp = Float64(fma(fma(Float64(y * y), 0.008333333333333333, 0.16666666666666666), Float64(y * y), 1.0) * y);
	elseif (x <= 7.4e+119)
		tmp = Float64(Float64(Float64(fma(-0.027777777777777776, Float64(x * x), 0.16666666666666666) * y) * y) * y);
	else
		tmp = Float64(fma(-0.16666666666666666, Float64(Float64(-x) * x), 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.75e+63], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 7.4e+119], N[(N[(N[(N[(-0.027777777777777776 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision], N[(N[(-0.16666666666666666 * N[((-x) * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.75000000000000015e63

   1. Initial program 90.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.9%

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

   if 1.75000000000000015e63 < x < 7.3999999999999999e119

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 2.2%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot {x}^{2} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 17.6%

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

   12. Taylor expanded in y around inf

       \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \cdot y \]
   13. Step-by-step derivation

   14. Applied rewrites 48.6%

       \[\leadsto \left(\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot y \]

   if 7.3999999999999999e119 < x

   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

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 58.3

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 13.8%

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

   10. Applied rewrites 29.2%

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

Alternative 11: 52.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 54:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+119}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \left(-x\right) \cdot x, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 54.0)
   (* (fma (* y y) 0.16666666666666666 1.0) y)
   (if (<= x 7.4e+119)
     (* (* (* (fma -0.027777777777777776 (* x x) 0.16666666666666666) y) y) y)
     (* (fma -0.16666666666666666 (* (- x) x) 1.0) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 54.0) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * y;
	} else if (x <= 7.4e+119) {
		tmp = ((fma(-0.027777777777777776, (x * x), 0.16666666666666666) * y) * y) * y;
	} else {
		tmp = fma(-0.16666666666666666, (-x * x), 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 54.0)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y);
	elseif (x <= 7.4e+119)
		tmp = Float64(Float64(Float64(fma(-0.027777777777777776, Float64(x * x), 0.16666666666666666) * y) * y) * y);
	else
		tmp = Float64(fma(-0.16666666666666666, Float64(Float64(-x) * x), 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 54.0], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 7.4e+119], N[(N[(N[(N[(-0.027777777777777776 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision], N[(N[(-0.16666666666666666 * N[((-x) * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 54

   1. Initial program 89.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 33.2%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot {x}^{2} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 59.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 56.3%

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

   if 54 < x < 7.3999999999999999e119

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 91.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 6.7%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot {x}^{2} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 11.4%

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

   12. Taylor expanded in y around inf

       \[\leadsto \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right) \cdot y \]
   13. Step-by-step derivation

   14. Applied rewrites 33.2%

       \[\leadsto \left(\left(\mathsf{fma}\left(-0.027777777777777776, x \cdot x, 0.16666666666666666\right) \cdot y\right) \cdot y\right) \cdot y \]

   if 7.3999999999999999e119 < x

   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

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 58.3

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 13.8%

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

   10. Applied rewrites 29.2%

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

Alternative 12: 51.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.9 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+119}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \left(-x\right) \cdot x, 1\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.9e+82)
   (* (fma (* y y) 0.16666666666666666 1.0) y)
   (if (<= x 7.4e+119)
     (* (* (* x x) -0.16666666666666666) y)
     (* (fma -0.16666666666666666 (* (- x) x) 1.0) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.9e+82) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * y;
	} else if (x <= 7.4e+119) {
		tmp = ((x * x) * -0.16666666666666666) * y;
	} else {
		tmp = fma(-0.16666666666666666, (-x * x), 1.0) * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.9e+82)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y);
	elseif (x <= 7.4e+119)
		tmp = Float64(Float64(Float64(x * x) * -0.16666666666666666) * y);
	else
		tmp = Float64(fma(-0.16666666666666666, Float64(Float64(-x) * x), 1.0) * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.9e+82], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 7.4e+119], N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * y), $MachinePrecision], N[(N[(-0.16666666666666666 * N[((-x) * x), $MachinePrecision] + 1.0), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.9000000000000001e82

   1. Initial program 90.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 31.5%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot {x}^{2} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 55.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 54.5%

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

   if 4.9000000000000001e82 < x < 7.3999999999999999e119

   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

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 63.4

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

   5. Applied rewrites 63.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 27.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 27.2%

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

   if 7.3999999999999999e119 < x

   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

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 58.3

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 13.8%

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

   10. Applied rewrites 29.2%

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

Alternative 13: 51.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{+192}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(y \cdot x\right) \cdot x, -0.16666666666666666, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.6e+192)
   (* (fma (* y y) 0.16666666666666666 1.0) y)
   (fma (* (* y x) x) -0.16666666666666666 y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.6e+192) {
		tmp = fma((y * y), 0.16666666666666666, 1.0) * y;
	} else {
		tmp = fma(((y * x) * x), -0.16666666666666666, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.6e+192)
		tmp = Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y);
	else
		tmp = fma(Float64(Float64(y * x) * x), -0.16666666666666666, y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.6e+192], N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666 + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.6000000000000002e192

   1. Initial program 91.3%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 28.6%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(1 + \left(\frac{-1}{6} \cdot {x}^{2} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{-1}{36} \cdot {x}^{2}\right)\right)\right) \cdot y \]
   10. Step-by-step derivation

   11. Applied rewrites 50.8%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 50.6%

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

   if 3.6000000000000002e192 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 20.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 21.0%

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

Alternative 14: 32.1% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{+42}:\\ \;\;\;\;1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.45e+42) (* 1.0 y) (* (* (* x x) -0.16666666666666666) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.45e+42) {
		tmp = 1.0 * y;
	} else {
		tmp = ((x * x) * -0.16666666666666666) * y;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= 2.45d+42) then
        tmp = 1.0d0 * y
    else
        tmp = ((x * x) * (-0.16666666666666666d0)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.45e+42) {
		tmp = 1.0 * y;
	} else {
		tmp = ((x * x) * -0.16666666666666666) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.45e+42:
		tmp = 1.0 * y
	else:
		tmp = ((x * x) * -0.16666666666666666) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.45e+42)
		tmp = Float64(1.0 * y);
	else
		tmp = Float64(Float64(Float64(x * x) * -0.16666666666666666) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.45e+42)
		tmp = 1.0 * y;
	else
		tmp = ((x * x) * -0.16666666666666666) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.45e+42], N[(1.0 * y), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4500000000000001e42

   1. Initial program 90.0%

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

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 31.0%

      \[\leadsto 1 \cdot y \]

   if 2.4500000000000001e42 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sin.f64 59.6

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 13.7%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 13.7%

       \[\leadsto \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 36.1% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(y \cdot x\right) \cdot x, -0.16666666666666666, y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (* (* y x) x) -0.16666666666666666 y))

C

double code(double x, double y) {
	return fma(((y * x) * x), -0.16666666666666666, y);
}

Julia

function code(x, y)
	return fma(Float64(Float64(y * x) * x), -0.16666666666666666, y)
end

Wolfram

code[x_, y_] := N[(N[(N[(y * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666 + y), $MachinePrecision]

Derivation

1. Initial program 92.2%

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

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 52.8

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

5. Applied rewrites 52.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 32.1%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 32.2%

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

Alternative 16: 27.8% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ 1 \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 1.0 y))

C

double code(double x, double y) {
	return 1.0 * 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 = 1.0d0 * y
end function

Java

public static double code(double x, double y) {
	return 1.0 * y;
}

Python

def code(x, y):
	return 1.0 * y

Julia

function code(x, y)
	return Float64(1.0 * y)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 * y;
end

Wolfram

code[x_, y_] := N[(1.0 * y), $MachinePrecision]

Derivation

1. Initial program 92.2%

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

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sin.f64 52.8

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

5. Applied rewrites 52.8%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 \cdot y \]
7. Step-by-step derivation

8. Applied rewrites 25.2%

   \[\leadsto 1 \cdot y \]
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 2025017 
(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))