Linear.Quaternion:$cexp from linear-1.19.1.3

Percentage Accurate: 99.8% → 99.8%

Time: 3.4s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (* x (/ (sin y) y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	x * ((sin(y)) / y)
END code

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (* x (/ (sin y) y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	x * ((sin(y)) / y)
END code

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9995:\\ \;\;\;\;\frac{x \cdot \sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (if (<= (/ (sin y) y) 0.9995)
  (/ (* x (sin y)) y)
  (* x (fma (* y y) -0.16666666666666666 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((sin(y) / y) <= 0.9995) {
		tmp = (x * sin(y)) / y;
	} else {
		tmp = x * fma((y * y), -0.16666666666666666, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.9995)
		tmp = Float64(Float64(x * sin(y)) / y);
	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[y], $MachinePrecision] / y), $MachinePrecision], 0.9995], N[(N[(x * N[Sin[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp = IF (((sin(y)) / y) <= (9995000000000000550670620214077644050121307373046875e-52)) THEN ((x * (sin(y))) / y) ELSE (x * (((y * y) * (-1666666666666666574148081281236954964697360992431640625e-55)) + (1))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99950000000000006

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 88.8%

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

   if 0.99950000000000006 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.5%

      \[\leadsto x \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 50.5%

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

Alternative 2: 63.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|y\right| \leq 26110.61450005739:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(-0.16666666666666666 \cdot \left|y\right|, \left|y\right|, 0.5\right) + 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(0.16666666666666666 \cdot \left|y\right|\right) \cdot \frac{\left|y\right|}{x}}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (if (<= (fabs y) 26110.61450005739)
  (* x (+ (fma (* -0.16666666666666666 (fabs y)) (fabs y) 0.5) 0.5))
  (/ 1.0 (* (* 0.16666666666666666 (fabs y)) (/ (fabs y) x)))))

C

double code(double x, double y) {
	double tmp;
	if (fabs(y) <= 26110.61450005739) {
		tmp = x * (fma((-0.16666666666666666 * fabs(y)), fabs(y), 0.5) + 0.5);
	} else {
		tmp = 1.0 / ((0.16666666666666666 * fabs(y)) * (fabs(y) / x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (abs(y) <= 26110.61450005739)
		tmp = Float64(x * Float64(fma(Float64(-0.16666666666666666 * abs(y)), abs(y), 0.5) + 0.5));
	else
		tmp = Float64(1.0 / Float64(Float64(0.16666666666666666 * abs(y)) * Float64(abs(y) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Abs[y], $MachinePrecision], 26110.61450005739], N[(x * N[(N[(N[(-0.16666666666666666 * N[Abs[y], $MachinePrecision]), $MachinePrecision] * N[Abs[y], $MachinePrecision] + 0.5), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(0.16666666666666666 * N[Abs[y], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp = IF ((abs(y)) <= (2611061450005738879553973674774169921875e-35)) THEN (x * (((((-1666666666666666574148081281236954964697360992431640625e-55) * (abs(y))) * (abs(y))) + (5e-1)) + (5e-1))) ELSE ((1) / (((1666666666666666574148081281236954964697360992431640625e-55) * (abs(y))) * ((abs(y)) / x))) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if y < 26110.614500057389

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.5%

      \[\leadsto x \cdot \left(1 + -0.16666666666666666 \cdot {y}^{2}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 50.5%

      \[\leadsto x \cdot \left(\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 0.5\right) + 0.5\right) \]

   if 26110.614500057389 < y

   1. Initial program 99.8%

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

   3. Applied rewrites 88.3%

      \[\leadsto \frac{1}{\frac{y}{\sin y \cdot x}} \]

   4. Taylor expanded in y around 0

      \[\leadsto \frac{1}{\frac{1}{6} \cdot \frac{{y}^{2}}{x} + \frac{1}{x}} \]
   5. Step-by-step derivation

   6. Applied rewrites 63.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(0.16666666666666666, \frac{{y}^{2}}{x}, \frac{1}{x}\right)} \]

   7. Taylor expanded in y around inf

      \[\leadsto \frac{1}{\frac{1}{6} \cdot \frac{{y}^{2}}{x}} \]
   8. Step-by-step derivation

   9. Applied rewrites 16.6%

      \[\leadsto \frac{1}{0.16666666666666666 \cdot \frac{{y}^{2}}{x}} \]

   11. Applied rewrites 16.6%

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

Alternative 3: 63.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 3.1115210104175464 \cdot 10^{-119}:\\ \;\;\;\;0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (if (<= (/ (sin y) y) 3.1115210104175464e-119) (* 0.0 1.0) (* x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((sin(y) / y) <= 3.1115210104175464e-119) {
		tmp = 0.0 * 1.0;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Fortran

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(y) / y) <= 3.1115210104175464d-119) then
        tmp = 0.0d0 * 1.0d0
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.sin(y) / y) <= 3.1115210104175464e-119) {
		tmp = 0.0 * 1.0;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.sin(y) / y) <= 3.1115210104175464e-119:
		tmp = 0.0 * 1.0
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 3.1115210104175464e-119)
		tmp = Float64(0.0 * 1.0);
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sin(y) / y) <= 3.1115210104175464e-119)
		tmp = 0.0 * 1.0;
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 3.1115210104175464e-119], N[(0.0 * 1.0), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp = IF (((sin(y)) / y) <= (311152101041754642290024565439875431740613962063175481148591194442996388678289957470708513147988336046540112239599631655494021672392081353289379760901846390424021238903116260518260409730045459215395629766697200463532105992136490754611180397706904128805760152687370442659154707830946630375823502845378243364393711090087890625e-442)) THEN ((0) * (1)) ELSE (x * (1)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 3.1115210104175464e-119

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

      \[\leadsto x \cdot 1 \]
   3. Step-by-step derivation

   4. Applied rewrites 51.2%

      \[\leadsto x \cdot 1 \]

   5. Taylor expanded in undef-var around zero

      \[\leadsto 0 \cdot 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 15.8%

      \[\leadsto 0 \cdot 1 \]

   if 3.1115210104175464e-119 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

      \[\leadsto x \cdot 1 \]
   3. Step-by-step derivation

   4. Applied rewrites 51.2%

      \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 63.2% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (/ 1.0 (fma (* 0.16666666666666666 y) (/ y x) (/ 1.0 x))))

C

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

Julia

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

Wolfram

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

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	(1) / ((((1666666666666666574148081281236954964697360992431640625e-55) * y) * (y / x)) + ((1) / x))
END code

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 88.3%

   \[\leadsto \frac{1}{\frac{y}{\sin y \cdot x}} \]

4. Taylor expanded in y around 0

   \[\leadsto \frac{1}{\frac{1}{6} \cdot \frac{{y}^{2}}{x} + \frac{1}{x}} \]
5. Step-by-step derivation

6. Applied rewrites 63.2%

   \[\leadsto \frac{1}{\mathsf{fma}\left(0.16666666666666666, \frac{{y}^{2}}{x}, \frac{1}{x}\right)} \]

8. Applied rewrites 63.2%

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

Alternative 5: 62.4% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (/ 1.0 (/ (fma (* y y) 0.16666666666666666 1.0) x)))

C

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

Julia

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

Wolfram

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

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	(1) / ((((y * y) * (1666666666666666574148081281236954964697360992431640625e-55)) + (1)) / x)
END code

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 88.3%

   \[\leadsto \frac{1}{\frac{y}{\sin y \cdot x}} \]

4. Taylor expanded in y around 0

   \[\leadsto \frac{1}{\frac{1}{6} \cdot \frac{{y}^{2}}{x} + \frac{1}{x}} \]
5. Step-by-step derivation

6. Applied rewrites 63.2%

   \[\leadsto \frac{1}{\mathsf{fma}\left(0.16666666666666666, \frac{{y}^{2}}{x}, \frac{1}{x}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 63.2%

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

Alternative 6: 51.2% accurate, 10.3× speedup

Math

\[x \cdot 1 \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	x * (1)
END code

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in y around 0

   \[\leadsto x \cdot 1 \]
3. Step-by-step derivation

4. Applied rewrites 51.2%

   \[\leadsto x \cdot 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2026092 
(FPCore (x y)
  :name "Linear.Quaternion:$cexp from linear-1.19.1.3"
  :precision binary64
  (* x (/ (sin y) y)))