Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 3.8s

Alternatives: 16

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 75.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 75.0

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \left({x}^{3} \cdot \frac{-1}{6}\right) \cdot \frac{\sinh y}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} \cdot \frac{-1}{6}\right) \cdot \frac{\sinh y}{y} \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 26.0

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

   7. Applied rewrites 26.0%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 74.8%

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

Alternative 3: 74.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(y) / y;
	double t_1 = Math.sin(x) * t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (((x * x) * x) * -0.16666666666666666) * t_0;
	} else if (t_1 <= 1.0) {
		tmp = Math.sin(x);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(y) / y
	t_1 = math.sin(x) * t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (((x * x) * x) * -0.16666666666666666) * t_0
	elif t_1 <= 1.0:
		tmp = math.sin(x)
	else:
		tmp = x * t_0
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(y) / y;
	t_1 = sin(x) * t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (((x * x) * x) * -0.16666666666666666) * t_0;
	elseif (t_1 <= 1.0)
		tmp = sin(x);
	else
		tmp = x * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[Sin[x], $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 75.0

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \left({x}^{3} \cdot \frac{-1}{6}\right) \cdot \frac{\sinh y}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} \cdot \frac{-1}{6}\right) \cdot \frac{\sinh y}{y} \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 26.0

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

   7. Applied rewrites 26.0%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lift-sin.f64 98.5

         \[\leadsto \sin x \]

   4. Applied rewrites 98.5%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 74.8%

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

Alternative 4: 62.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;\sin x \cdot t\_0 \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* (sin x) t_0) 5e-82)
     (* (* (fma -0.16666666666666666 (* x x) 1.0) x) t_0)
     (/ (* x (sinh y)) y))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((sin(x) * t_0) <= 5e-82) {
		tmp = (fma(-0.16666666666666666, (x * x), 1.0) * x) * t_0;
	} else {
		tmp = (x * sinh(y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(sin(x) * t_0) <= 5e-82)
		tmp = Float64(Float64(fma(-0.16666666666666666, Float64(x * x), 1.0) * x) * t_0);
	else
		tmp = Float64(Float64(x * sinh(y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision], 5e-82], N[(N[(N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 4.9999999999999998e-82

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 68.9

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

   4. Applied rewrites 68.9%

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

   if 4.9999999999999998e-82 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 53.7

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

   6. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{x \cdot \sinh y}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 62.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;\sin x \cdot t\_0 \leq -0.05:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot -0.16666666666666666\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* (sin x) t_0) -0.05)
     (* (* (* (* x x) x) -0.16666666666666666) t_0)
     (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((sin(x) * t_0) <= -0.05) {
		tmp = (((x * x) * x) * -0.16666666666666666) * t_0;
	} else {
		tmp = x * t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = sinh(y) / y
    if ((sin(x) * t_0) <= (-0.05d0)) then
        tmp = (((x * x) * x) * (-0.16666666666666666d0)) * t_0
    else
        tmp = x * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(y) / y;
	double tmp;
	if ((Math.sin(x) * t_0) <= -0.05) {
		tmp = (((x * x) * x) * -0.16666666666666666) * t_0;
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(y) / y
	tmp = 0
	if (math.sin(x) * t_0) <= -0.05:
		tmp = (((x * x) * x) * -0.16666666666666666) * t_0
	else:
		tmp = x * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(sin(x) * t_0) <= -0.05)
		tmp = Float64(Float64(Float64(Float64(x * x) * x) * -0.16666666666666666) * t_0);
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(y) / y;
	tmp = 0.0;
	if ((sin(x) * t_0) <= -0.05)
		tmp = (((x * x) * x) * -0.16666666666666666) * t_0;
	else
		tmp = x * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision], -0.05], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * t$95$0), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 50.6

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \left({x}^{3} \cdot \frac{-1}{6}\right) \cdot \frac{\sinh y}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \left({x}^{3} \cdot \frac{-1}{6}\right) \cdot \frac{\sinh y}{y} \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 18.0

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

   7. Applied rewrites 18.0%

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

   if -0.050000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 70.5%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 55.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(-0.16666666666666666 \cdot x\right), x, x\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 5e-82)
   (*
    (fma (* x (* -0.16666666666666666 x)) x x)
    (fma (* y y) 0.16666666666666666 1.0))
   (/ (* x (sinh y)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 5e-82) {
		tmp = fma((x * (-0.16666666666666666 * x)), x, x) * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = (x * sinh(y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 5e-82)
		tmp = Float64(fma(Float64(x * Float64(-0.16666666666666666 * x)), x, x) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(Float64(x * sinh(y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 5e-82], N[(N[(N[(x * N[(-0.16666666666666666 * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 4.9999999999999998e-82

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 68.9

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

   4. Applied rewrites 68.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. pow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-*r* N/A

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

      12. *-rgt-identity N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 68.9

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

   6. Applied rewrites 68.9%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 57.3

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

   9. Applied rewrites 57.3%

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

   if 4.9999999999999998e-82 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 53.7

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

   6. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\frac{x \cdot \sinh y}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 51.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;\sin x \cdot t\_0 \leq -0.05:\\ \;\;\;\;{x}^{7} \cdot -0.0001984126984126984\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* (sin x) t_0) -0.05)
     (* (pow x 7.0) -0.0001984126984126984)
     (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((sin(x) * t_0) <= -0.05) {
		tmp = pow(x, 7.0) * -0.0001984126984126984;
	} else {
		tmp = x * t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = sinh(y) / y
    if ((sin(x) * t_0) <= (-0.05d0)) then
        tmp = (x ** 7.0d0) * (-0.0001984126984126984d0)
    else
        tmp = x * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(y) / y;
	double tmp;
	if ((Math.sin(x) * t_0) <= -0.05) {
		tmp = Math.pow(x, 7.0) * -0.0001984126984126984;
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(y) / y
	tmp = 0
	if (math.sin(x) * t_0) <= -0.05:
		tmp = math.pow(x, 7.0) * -0.0001984126984126984
	else:
		tmp = x * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (Float64(sin(x) * t_0) <= -0.05)
		tmp = Float64((x ^ 7.0) * -0.0001984126984126984);
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(y) / y;
	tmp = 0.0;
	if ((sin(x) * t_0) <= -0.05)
		tmp = (x ^ 7.0) * -0.0001984126984126984;
	else
		tmp = x * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision], -0.05], N[(N[Power[x, 7.0], $MachinePrecision] * -0.0001984126984126984), $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lift-sin.f64 34.7

         \[\leadsto \sin x \]

   4. Applied rewrites 34.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 16.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{-1}{5040} \cdot {x}^{\color{blue}{7}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {x}^{7} \cdot \frac{-1}{5040} \]

      2. lower-*.f64 N/A

         \[\leadsto {x}^{7} \cdot \frac{-1}{5040} \]

      3. lower-pow.f64 16.2

         \[\leadsto {x}^{7} \cdot -0.0001984126984126984 \]

   10. Applied rewrites 16.2%

       \[\leadsto {x}^{7} \cdot -0.0001984126984126984 \]

   if -0.050000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 70.5%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 50.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin x \cdot \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq -0.05:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\right) \cdot y}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{+192}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sin x) (/ (sinh y) y))))
   (if (<= t_0 -0.05)
     (/ (* (* (fma (* x x) -0.16666666666666666 1.0) x) y) y)
     (if (<= t_0 1e+192)
       (* x (fma (* y y) 0.16666666666666666 1.0))
       (/ (* x (sinh y)) y)))))

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -0.05) {
		tmp = ((fma((x * x), -0.16666666666666666, 1.0) * x) * y) / y;
	} else if (t_0 <= 1e+192) {
		tmp = x * fma((y * y), 0.16666666666666666, 1.0);
	} else {
		tmp = (x * sinh(y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(sin(x) * Float64(sinh(y) / y))
	tmp = 0.0
	if (t_0 <= -0.05)
		tmp = Float64(Float64(Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x) * y) / y);
	elseif (t_0 <= 1e+192)
		tmp = Float64(x * fma(Float64(y * y), 0.16666666666666666, 1.0));
	else
		tmp = Float64(Float64(x * sinh(y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.05], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$0, 1e+192], N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Sinh[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 50.6

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

   4. Applied rewrites 50.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. pow2 N/A

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

      12. cube-mult N/A

          \[\leadsto \left({x}^{3} \cdot \frac{-1}{6} + x \cdot 1\right) \cdot \frac{\sinh y}{y} \]

      13. unpow3 N/A

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

      14. pow2 N/A

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

      15. associate-*l* N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. lower-*.f64 50.6

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

   6. Applied rewrites 50.6%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 13.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

   11. Applied rewrites 16.2%

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

   if -0.050000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 1.00000000000000004e192

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 67.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 66.9

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

   7. Applied rewrites 66.9%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 66.9

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

   9. Applied rewrites 66.9%

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

   10. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 66.9

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

   12. Applied rewrites 66.9%

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

   if 1.00000000000000004e192 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 75.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sinh.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sinh.f64 75.1

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

   6. Applied rewrites 75.1%

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

Alternative 9: 50.0% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.005)
   (/ (* (* (fma (* x x) -0.16666666666666666 1.0) x) y) y)
   (* x (/ (sinh y) y))))

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.005], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0050000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 26.9

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

   4. Applied rewrites 26.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. pow2 N/A

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

      12. cube-mult N/A

          \[\leadsto \left({x}^{3} \cdot \frac{-1}{6} + x \cdot 1\right) \cdot \frac{\sinh y}{y} \]

      13. unpow3 N/A

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

      14. pow2 N/A

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

      15. associate-*l* N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. lower-*.f64 26.9

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

   6. Applied rewrites 26.9%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 18.1%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

   11. Applied rewrites 22.6%

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

   if -0.0050000000000000001 < (sin.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 75.7%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 47.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq -0.05:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\right) \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y}{y} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) -0.05)
   (/ (* (* (fma (* x x) -0.16666666666666666 1.0) x) y) y)
   (* (/ (* (fma (* y y) 0.16666666666666666 1.0) y) y) x)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= -0.05) {
		tmp = ((fma((x * x), -0.16666666666666666, 1.0) * x) * y) / y;
	} else {
		tmp = ((fma((y * y), 0.16666666666666666, 1.0) * y) / y) * x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= -0.05)
		tmp = Float64(Float64(Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x) * y) / y);
	else
		tmp = Float64(Float64(Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y) / y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < -0.050000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 50.6

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

   4. Applied rewrites 50.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. pow2 N/A

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

      12. cube-mult N/A

          \[\leadsto \left({x}^{3} \cdot \frac{-1}{6} + x \cdot 1\right) \cdot \frac{\sinh y}{y} \]

      13. unpow3 N/A

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

      14. pow2 N/A

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

      15. associate-*l* N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. lower-*.f64 50.6

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

   6. Applied rewrites 50.6%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 13.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

   11. Applied rewrites 16.2%

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

   if -0.050000000000000003 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 62.2

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

   7. Applied rewrites 62.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 62.2

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

      4. lift-fma.f64 N/A

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

      5. distribute-lft1-in N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 62.2

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

   9. Applied rewrites 62.2%

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

Alternative 11: 45.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right) \cdot y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 5e-82)
   (* (fma (* x x) -0.16666666666666666 1.0) x)
   (/ (* x (* (fma (* y y) 0.16666666666666666 1.0) y)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 5e-82) {
		tmp = fma((x * x), -0.16666666666666666, 1.0) * x;
	} else {
		tmp = (x * (fma((y * y), 0.16666666666666666, 1.0) * y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 5e-82)
		tmp = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x);
	else
		tmp = Float64(Float64(x * Float64(fma(Float64(y * y), 0.16666666666666666, 1.0) * y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 5e-82], N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 4.9999999999999998e-82

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lift-sin.f64 59.2

         \[\leadsto \sin x \]

   4. Applied rewrites 59.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 47.3%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 45.2

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

   10. Applied rewrites 45.2%

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

   if 4.9999999999999998e-82 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 54.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 41.7

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

   7. Applied rewrites 41.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   9. Applied rewrites 41.8%

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

Alternative 12: 43.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(\left(y \cdot y\right) \cdot y\right) \cdot 0.16666666666666666\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 5e-14)
   (* (fma (* x x) -0.16666666666666666 1.0) x)
   (/ (* x (* (* (* y y) y) 0.16666666666666666)) y)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 5e-14) {
		tmp = fma((x * x), -0.16666666666666666, 1.0) * x;
	} else {
		tmp = (x * (((y * y) * y) * 0.16666666666666666)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 5e-14)
		tmp = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x);
	else
		tmp = Float64(Float64(x * Float64(Float64(Float64(y * y) * y) * 0.16666666666666666)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 5e-14], N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 5.0000000000000002e-14

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lift-sin.f64 60.6

         \[\leadsto \sin x \]

   4. Applied rewrites 60.6%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 49.1%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 47.1

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

   10. Applied rewrites 47.1%

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

   if 5.0000000000000002e-14 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 38.2

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

   7. Applied rewrites 38.2%

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

   8. Taylor expanded in y around inf

      \[\leadsto x \cdot \frac{\frac{1}{6} \cdot \color{blue}{{y}^{3}}}{y} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot \frac{{y}^{3} \cdot \frac{1}{6}}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \frac{{y}^{3} \cdot \frac{1}{6}}{y} \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 37.1

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

   10. Applied rewrites 37.1%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 38.0

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

       6. *-commutative 38.0

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

       7. associate-*l* 38.0

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

       8. associate-*l* 38.0

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

   12. Applied rewrites 38.0%

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

Alternative 13: 43.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \cdot \frac{\sinh y}{y} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\left(y \cdot y\right) \cdot \left(0.16666666666666666 \cdot y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 5e-14)
   (* (fma (* x x) -0.16666666666666666 1.0) x)
   (* x (/ (* (* y y) (* 0.16666666666666666 y)) y))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(x) * (sinh(y) / y)) <= 5e-14) {
		tmp = fma((x * x), -0.16666666666666666, 1.0) * x;
	} else {
		tmp = x * (((y * y) * (0.16666666666666666 * y)) / y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 5e-14)
		tmp = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x);
	else
		tmp = Float64(x * Float64(Float64(Float64(y * y) * Float64(0.16666666666666666 * y)) / y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 5e-14], N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(x * N[(N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y)) < 5.0000000000000002e-14

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lift-sin.f64 60.6

         \[\leadsto \sin x \]

   4. Applied rewrites 60.6%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 49.1%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 47.1

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

   10. Applied rewrites 47.1%

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

   if 5.0000000000000002e-14 < (*.f64 (sin.f64 x) (/.f64 (sinh.f64 y) y))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 51.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 38.2

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

   7. Applied rewrites 38.2%

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

   8. Taylor expanded in y around inf

      \[\leadsto x \cdot \frac{\frac{1}{6} \cdot \color{blue}{{y}^{3}}}{y} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot \frac{{y}^{3} \cdot \frac{1}{6}}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot \frac{{y}^{3} \cdot \frac{1}{6}}{y} \]

      3. unpow3 N/A

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

      4. pow2 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 37.1

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

   10. Applied rewrites 37.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 37.1

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

   12. Applied rewrites 37.1%

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

Alternative 14: 43.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq -0.005:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) -0.005)
   (* (fma (* x x) -0.16666666666666666 1.0) x)
   (* x (fma (* y y) 0.16666666666666666 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= -0.005) {
		tmp = fma((x * x), -0.16666666666666666, 1.0) * x;
	} else {
		tmp = x * fma((y * y), 0.16666666666666666, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= -0.005)
		tmp = Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x);
	else
		tmp = Float64(x * fma(Float64(y * y), 0.16666666666666666, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], -0.005], N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < -0.0050000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lift-sin.f64 51.1

         \[\leadsto \sin x \]

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 23.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 18.1

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

   10. Applied rewrites 18.1%

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

   if -0.0050000000000000001 < (sin.f64 x)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 75.7%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 63.3

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

   7. Applied rewrites 63.3%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 63.3

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

   9. Applied rewrites 63.3%

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

   10. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 57.4

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

   12. Applied rewrites 57.4%

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

Alternative 15: 34.9% accurate, 4.3× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (fma (* x x) -0.16666666666666666 1.0) x))

C

double code(double x, double y) {
	return fma((x * x), -0.16666666666666666, 1.0) * x;
}

Julia

function code(x, y)
	return Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x)
end

Wolfram

code[x_, y_] := N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around 0

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

   1. lift-sin.f64 51.3

      \[\leadsto \sin x \]

4. Applied rewrites 51.3%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 37.1%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 34.9

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

10. Applied rewrites 34.9%

    \[\leadsto \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x \]
11. Add Preprocessing

Alternative 16: 27.2% accurate, 51.3× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in y around 0

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

   1. lift-sin.f64 51.3

      \[\leadsto \sin x \]

4. Applied rewrites 51.3%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 37.1%

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

8. Taylor expanded in x around 0

   \[\leadsto x \]
9. Step-by-step derivation

10. Applied rewrites 27.2%

    \[\leadsto x \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025112 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))