Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

Alternatives: 14

Speedup: 1.0×

• 49.8% 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: 74.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	t_0 = fma(Float64(y * y), 0.16666666666666666, 1.0)
	t_1 = Float64(sin(x) * Float64(sinh(y) / y))
	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) * t_0);
	else
		tmp = Float64(Float64(x * sinh(y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $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] * t$95$0), $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)) < -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 24.6

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

   7. Applied rewrites 24.6%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 23.0

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

   10. Applied rewrites 23.0%

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

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

   1. Initial program 100.0%

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

   2. 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 99.2

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

   4. Applied rewrites 99.2%

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

   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 74.5%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]
   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 74.5

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

   6. Applied rewrites 74.5%

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

Alternative 3: 73.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = sin(x) * (sinh(y) / y);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (((x * x) * x) * -0.16666666666666666) * fma((y * y), 0.16666666666666666, 1.0);
	} else if (t_0 <= 1.0) {
		tmp = sin(x);
	} 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 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(x * x) * x) * -0.16666666666666666) * fma(Float64(y * y), 0.16666666666666666, 1.0));
	elseif (t_0 <= 1.0)
		tmp = sin(x);
	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, (-Infinity)], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[Sin[x], $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)) < -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 24.6

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

   7. Applied rewrites 24.6%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 23.0

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

   10. Applied rewrites 23.0%

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

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

   1. Initial program 100.0%

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

   2. 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 74.5%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]
   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 74.5

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

   6. Applied rewrites 74.5%

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

Alternative 4: 63.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* (sin x) t_0) 0.05)
     (* (* (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) <= 0.05) {
		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) <= 0.05)
		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], 0.05], 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)) < 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 70.2

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

   4. Applied rewrites 70.2%

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

   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 51.7%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]
   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 51.7

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

   6. Applied rewrites 51.7%

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

Alternative 5: 50.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;\sin x \cdot t\_0 \leq -0.04:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot -0.16666666666666666\right) \cdot \mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* (sin x) t_0) -0.04)
     (*
      (* (* (* x x) x) -0.16666666666666666)
      (fma (* y y) 0.16666666666666666 1.0))
     (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((sin(x) * t_0) <= -0.04) {
		tmp = (((x * x) * x) * -0.16666666666666666) * 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)
	tmp = 0.0
	if (Float64(sin(x) * t_0) <= -0.04)
		tmp = Float64(Float64(Float64(Float64(x * x) * x) * -0.16666666666666666) * 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]}, If[LessEqual[N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision], -0.04], N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $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)) < -0.0400000000000000008

   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 51.1

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

   4. Applied rewrites 51.1%

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

   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 17.2

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

   7. Applied rewrites 17.2%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 16.1

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

   10. Applied rewrites 16.1%

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

   if -0.0400000000000000008 < (*.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 70.1%

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

Alternative 6: 49.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;\sin x \cdot t\_0 \leq -0.04:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y)))
   (if (<= (* (sin x) t_0) -0.04)
     (* (fma (* -0.16666666666666666 x) x 1.0) x)
     (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if ((sin(x) * t_0) <= -0.04) {
		tmp = fma((-0.16666666666666666 * x), x, 1.0) * x;
	} 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) <= -0.04)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * x);
	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]}, If[LessEqual[N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision], -0.04], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $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)) < -0.0400000000000000008

   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 34.2

         \[\leadsto \sin x \]

   4. Applied rewrites 34.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 15.4

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

   7. Applied rewrites 15.4%

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

   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. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 12.9

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

   10. Applied rewrites 12.9%

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

   if -0.0400000000000000008 < (*.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 70.1%

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

Alternative 7: 48.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 0.05)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * x);
	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], 0.05], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $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)) < 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 61.1

         \[\leadsto \sin x \]

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 48.8

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

   7. Applied rewrites 48.8%

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

   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. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 47.3

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

   10. Applied rewrites 47.3%

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

   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 51.7%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{y} \]
   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 51.7

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

   6. Applied rewrites 51.7%

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

Alternative 8: 43.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 0.05)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * x);
	else
		tmp = Float64(Float64(x * Float64(fma(y, Float64(0.16666666666666666 * y), 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], 0.05], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[(N[(y * N[(0.16666666666666666 * y), $MachinePrecision] + 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)) < 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 61.1

         \[\leadsto \sin x \]

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 48.8

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

   7. Applied rewrites 48.8%

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

   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. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 47.3

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

   10. Applied rewrites 47.3%

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

   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 51.7%

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

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      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 36.4

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

   7. Applied rewrites 36.4%

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.16666666666666666, y, y\right)}}{y} \]
   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 37.4%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 37.4

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

   11. Applied rewrites 37.4%

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

Alternative 9: 43.6% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 0.05)
   (* (fma (* -0.16666666666666666 x) x 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)) <= 0.05) {
		tmp = fma((-0.16666666666666666 * x), x, 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)) <= 0.05)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * x);
	else
		tmp = Float64(Float64(x * Float64(Float64(Float64(y * y) * 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], 0.05], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[(N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / y), $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 61.1

         \[\leadsto \sin x \]

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 48.8

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

   7. Applied rewrites 48.8%

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

   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. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 47.3

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

   10. Applied rewrites 47.3%

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

   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 51.7%

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

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      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 36.4

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

   7. Applied rewrites 36.4%

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.16666666666666666, y, y\right)}}{y} \]
   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 37.4%

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

   10. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 37.4

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

   12. Applied rewrites 37.4%

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

Alternative 10: 43.2% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* (sin x) (/ (sinh y) y)) 0.05)
   (* (fma (* -0.16666666666666666 x) x 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)) <= 0.05) {
		tmp = fma((-0.16666666666666666 * x), x, 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)) <= 0.05)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * x);
	else
		tmp = Float64(x * Float64(Float64(y * Float64(Float64(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], 0.05], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(x * N[(N[(y * N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $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 61.1

         \[\leadsto \sin x \]

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 48.8

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

   7. Applied rewrites 48.8%

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

   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. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 47.3

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

   10. Applied rewrites 47.3%

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

   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 51.7%

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

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      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 36.4

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

   7. Applied rewrites 36.4%

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.16666666666666666, y, y\right)}}{y} \]
   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 36.4

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

   9. Applied rewrites 36.4%

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

   10. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. unpow3 N/A

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

       3. pow2 N/A

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

       4. associate-*r* N/A

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

       5. pow2 N/A

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

       6. associate-*l* N/A

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

       7. associate-*l* N/A

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

       8. pow2 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

       13. pow2 N/A

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

       14. lift-*.f64 36.4

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

   12. Applied rewrites 36.4%

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

Alternative 11: 41.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(x) * Float64(sinh(y) / y)) <= 0.05)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, 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[(N[Sin[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.05], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 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 (*.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 61.1

         \[\leadsto \sin x \]

   4. Applied rewrites 61.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 48.8

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

   7. Applied rewrites 48.8%

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

   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. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 47.3

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

   10. Applied rewrites 47.3%

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

   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 51.7%

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

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      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 36.4

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

   7. Applied rewrites 36.4%

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.16666666666666666, y, y\right)}}{y} \]
   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 36.4

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

   9. Applied rewrites 36.4%

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

   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

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

       2. pow2 N/A

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

       3. +-commutative N/A

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

       4. lift-fma.f64 N/A

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

       5. lift-*.f64 30.6

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

   12. Applied rewrites 30.6%

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

Alternative 12: 33.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq 0.033:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 \cdot y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) 0.033)
   (* (fma (* -0.16666666666666666 x) x 1.0) x)
   (/ (* x (* 1.0 y)) y)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= 0.033) {
		tmp = fma((-0.16666666666666666 * x), x, 1.0) * x;
	} else {
		tmp = (x * (1.0 * y)) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= 0.033)
		tmp = Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * x);
	else
		tmp = Float64(Float64(x * Float64(1.0 * y)) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], 0.033], N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[(1.0 * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 0.033000000000000002

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 41.3

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

   7. Applied rewrites 41.3%

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

   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. pow2 N/A

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

      3. associate-*r* N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 39.9

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

   10. Applied rewrites 39.9%

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

   if 0.033000000000000002 < (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 27.3%

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

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      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 21.3

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

   7. Applied rewrites 21.3%

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

      1. lift-*.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 22.7%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 14.2%

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

Alternative 13: 29.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin x \leq 10^{-52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(1 \cdot y\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sin x) 1e-52) x (/ (* x (* 1.0 y)) y)))

C

double code(double x, double y) {
	double tmp;
	if (sin(x) <= 1e-52) {
		tmp = x;
	} else {
		tmp = (x * (1.0 * y)) / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (sin(x) <= 1d-52) then
        tmp = x
    else
        tmp = (x * (1.0d0 * y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sin(x) <= 1e-52) {
		tmp = x;
	} else {
		tmp = (x * (1.0 * y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sin(x) <= 1e-52:
		tmp = x
	else:
		tmp = (x * (1.0 * y)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sin(x) <= 1e-52)
		tmp = x;
	else
		tmp = Float64(Float64(x * Float64(1.0 * y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sin(x) <= 1e-52)
		tmp = x;
	else
		tmp = (x * (1.0 * y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sin[x], $MachinePrecision], 1e-52], x, N[(N[(x * N[(1.0 * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 x) < 1e-52

   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 51.2

         \[\leadsto \sin x \]

   4. Applied rewrites 51.2%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 40.8

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

   7. Applied rewrites 40.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 34.2%

       \[\leadsto x \]

   if 1e-52 < (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 37.9%

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

   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. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. +-commutative N/A

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

      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 30.0

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

   7. Applied rewrites 30.0%

      \[\leadsto x \cdot \frac{\color{blue}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot 0.16666666666666666, y, y\right)}}{y} \]
   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 31.0%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 19.0%

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

Alternative 14: 26.9% 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 50.8

      \[\leadsto \sin x \]

4. Applied rewrites 50.8%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 36.4

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

7. Applied rewrites 36.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 26.9%

    \[\leadsto x \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025114 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))