Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.0% → 99.7%

Time: 8.4s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.3 \cdot 10^{-27}:\\ \;\;\;\;\frac{x\_m}{z} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (* x_s (if (<= x_m 2.3e-27) (* (/ x_m z) t_0) (/ (* t_0 x_m) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (x_m <= 2.3e-27) {
		tmp = (x_m / z) * t_0;
	} else {
		tmp = (t_0 * x_m) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (x_m <= 2.3d-27) then
        tmp = (x_m / z) * t_0
    else
        tmp = (t_0 * x_m) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (x_m <= 2.3e-27) {
		tmp = (x_m / z) * t_0;
	} else {
		tmp = (t_0 * x_m) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if x_m <= 2.3e-27:
		tmp = (x_m / z) * t_0
	else:
		tmp = (t_0 * x_m) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (x_m <= 2.3e-27)
		tmp = Float64(Float64(x_m / z) * t_0);
	else
		tmp = Float64(Float64(t_0 * x_m) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (x_m <= 2.3e-27)
		tmp = (x_m / z) * t_0;
	else
		tmp = (t_0 * x_m) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 2.3e-27], N[(N[(x$95$m / z), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(t$95$0 * x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2999999999999999e-27

   1. Initial program 92.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.4

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

   4. Applied rewrites 97.4%

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

   if 2.2999999999999999e-27 < x

   1. Initial program 99.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y}{y} \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\frac{\sin y}{y} \cdot x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)}{z} \cdot x\_m\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-318}:\\ \;\;\;\;\frac{x\_m}{z \cdot z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* (/ (sin y) y) x_m) z)))
   (*
    x_s
    (if (<= t_0 -5e-125)
      (* (/ (fma (* y y) -0.16666666666666666 1.0) z) x_m)
      (if (<= t_0 5e-318) (* (/ x_m (* z z)) z) (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = ((sin(y) / y) * x_m) / z;
	double tmp;
	if (t_0 <= -5e-125) {
		tmp = (fma((y * y), -0.16666666666666666, 1.0) / z) * x_m;
	} else if (t_0 <= 5e-318) {
		tmp = (x_m / (z * z)) * z;
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(Float64(sin(y) / y) * x_m) / z)
	tmp = 0.0
	if (t_0 <= -5e-125)
		tmp = Float64(Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) / z) * x_m);
	elseif (t_0 <= 5e-318)
		tmp = Float64(Float64(x_m / Float64(z * z)) * z);
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -5e-125], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[t$95$0, 5e-318], N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -4.99999999999999967e-125

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 89.3

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

   4. Applied rewrites 89.3%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x \cdot {y}^{2}}{z} \cdot \frac{-1}{6}} + \frac{x}{z} \]

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. associate-*r/ N/A

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

      11. associate-*l/ N/A

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

      12. associate-*r/ N/A

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

      13. distribute-lft1-in N/A

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

      14. +-commutative N/A

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

      15. associate-*r/ N/A

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

      16. *-rgt-identity N/A

          \[\leadsto x \cdot \frac{\color{blue}{1 + \frac{-1}{6} \cdot {y}^{2}}}{z} \]

      17. lower-/.f64 N/A

          \[\leadsto x \cdot \color{blue}{\frac{1 + \frac{-1}{6} \cdot {y}^{2}}{z}} \]

   7. Applied rewrites 63.8%

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

   if -4.99999999999999967e-125 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 4.9999987e-318

   1. Initial program 87.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 49.2

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.1%

      \[\leadsto \frac{-1}{z} \cdot \color{blue}{\left(-x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 49.1%

      \[\leadsto \frac{\frac{-1}{z}}{\color{blue}{\frac{-1}{x}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 56.3%

       \[\leadsto \frac{-x}{-z \cdot z} \cdot \color{blue}{z} \]

   if 4.9999987e-318 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 56.7

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq -5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)}{z} \cdot x\\ \mathbf{elif}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq 5 \cdot 10^{-318}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_0 \cdot x\_m}{z} \leq -2 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sin y}{z \cdot y} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (*
    x_s
    (if (<= (/ (* t_0 x_m) z) -2e-31)
      (* (/ (sin y) (* z y)) x_m)
      (* (/ x_m z) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (((t_0 * x_m) / z) <= -2e-31) {
		tmp = (sin(y) / (z * y)) * x_m;
	} else {
		tmp = (x_m / z) * t_0;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (((t_0 * x_m) / z) <= (-2d-31)) then
        tmp = (sin(y) / (z * y)) * x_m
    else
        tmp = (x_m / z) * t_0
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (((t_0 * x_m) / z) <= -2e-31) {
		tmp = (Math.sin(y) / (z * y)) * x_m;
	} else {
		tmp = (x_m / z) * t_0;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if ((t_0 * x_m) / z) <= -2e-31:
		tmp = (math.sin(y) / (z * y)) * x_m
	else:
		tmp = (x_m / z) * t_0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (Float64(Float64(t_0 * x_m) / z) <= -2e-31)
		tmp = Float64(Float64(sin(y) / Float64(z * y)) * x_m);
	else
		tmp = Float64(Float64(x_m / z) * t_0);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (((t_0 * x_m) / z) <= -2e-31)
		tmp = (sin(y) / (z * y)) * x_m;
	else
		tmp = (x_m / z) * t_0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(t$95$0 * x$95$m), $MachinePrecision] / z), $MachinePrecision], -2e-31], N[(N[(N[Sin[y], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * t$95$0), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -2e-31

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.6

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

   4. Applied rewrites 87.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(z\right)}} \cdot \frac{\sin y}{y} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(z\right)} \cdot \color{blue}{\frac{\sin y}{y}} \]

      6. frac-times N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \sin y}{\left(\mathsf{neg}\left(z\right)\right) \cdot y}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \sin y}{\color{blue}{\mathsf{neg}\left(z \cdot y\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \sin y}{\mathsf{neg}\left(\color{blue}{z \cdot y}\right)} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\sin y}{\mathsf{neg}\left(z \cdot y\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\sin y}{\mathsf{neg}\left(z \cdot y\right)}} \]

      11. lower-neg.f64 N/A

          \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{\sin y}{\mathsf{neg}\left(z \cdot y\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(-x\right) \cdot \color{blue}{\frac{\sin y}{\mathsf{neg}\left(z \cdot y\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(-x\right) \cdot \frac{\sin y}{\mathsf{neg}\left(\color{blue}{z \cdot y}\right)} \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \left(-x\right) \cdot \frac{\sin y}{\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot y}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(-x\right) \cdot \frac{\sin y}{\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot y}} \]

      16. lower-neg.f64 92.6

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

   6. Applied rewrites 92.6%

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

   if -2e-31 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 93.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.6

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

   4. Applied rewrites 95.6%

      \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \frac{x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq -2 \cdot 10^{-31}:\\ \;\;\;\;\frac{\sin y}{z \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x\_m}{z} \leq 5 \cdot 10^{-318}:\\ \;\;\;\;\frac{x\_m}{z \cdot z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* (/ (sin y) y) x_m) z) 5e-318)
    (* (/ x_m (* z z)) z)
    (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((((sin(y) / y) * x_m) / z) <= 5e-318) {
		tmp = (x_m / (z * z)) * z;
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((((sin(y) / y) * x_m) / z) <= 5d-318) then
        tmp = (x_m / (z * z)) * z
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((((Math.sin(y) / y) * x_m) / z) <= 5e-318) {
		tmp = (x_m / (z * z)) * z;
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (((math.sin(y) / y) * x_m) / z) <= 5e-318:
		tmp = (x_m / (z * z)) * z
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(sin(y) / y) * x_m) / z) <= 5e-318)
		tmp = Float64(Float64(x_m / Float64(z * z)) * z);
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((((sin(y) / y) * x_m) / z) <= 5e-318)
		tmp = (x_m / (z * z)) * z;
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], 5e-318], N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 4.9999987e-318

   1. Initial program 92.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 53.5

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 53.4%

      \[\leadsto \frac{-1}{z} \cdot \color{blue}{\left(-x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 53.3%

      \[\leadsto \frac{\frac{-1}{z}}{\color{blue}{\frac{-1}{x}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 49.6%

       \[\leadsto \frac{-x}{-z \cdot z} \cdot \color{blue}{z} \]

   if 4.9999987e-318 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 56.7

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq 5 \cdot 10^{-318}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x\_m}{z} \leq 0:\\ \;\;\;\;\frac{y \cdot x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* (/ (sin y) y) x_m) z) 0.0) (/ (* y x_m) (* z y)) (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((((sin(y) / y) * x_m) / z) <= 0.0) {
		tmp = (y * x_m) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((((sin(y) / y) * x_m) / z) <= 0.0d0) then
        tmp = (y * x_m) / (z * y)
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((((Math.sin(y) / y) * x_m) / z) <= 0.0) {
		tmp = (y * x_m) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (((math.sin(y) / y) * x_m) / z) <= 0.0:
		tmp = (y * x_m) / (z * y)
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(sin(y) / y) * x_m) / z) <= 0.0)
		tmp = Float64(Float64(y * x_m) / Float64(z * y));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((((sin(y) / y) * x_m) / z) <= 0.0)
		tmp = (y * x_m) / (z * y);
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], 0.0], N[(N[(y * x$95$m), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -0.0

   1. Initial program 92.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 89.9

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

   4. Applied rewrites 89.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 51.6

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

   7. Applied rewrites 51.6%

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

   if -0.0 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 56.7

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq 0:\\ \;\;\;\;\frac{y \cdot x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z \cdot y} \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 0.005)
    (*
     (fma (fma (* y y) 0.008333333333333333 -0.16666666666666666) (* y y) 1.0)
     (/ x_m z))
    (* (/ (sin y) (* z y)) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 0.005) {
		tmp = fma(fma((y * y), 0.008333333333333333, -0.16666666666666666), (y * y), 1.0) * (x_m / z);
	} else {
		tmp = (sin(y) / (z * y)) * x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 0.005)
		tmp = Float64(fma(fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), Float64(y * y), 1.0) * Float64(x_m / z));
	else
		tmp = Float64(Float64(sin(y) / Float64(z * y)) * x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 0.005], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 0.0050000000000000001

   1. Initial program 95.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 65.0

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

   5. Applied rewrites 65.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 66.0

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

   7. Applied rewrites 66.0%

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

   if 0.0050000000000000001 < y

   1. Initial program 90.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 86.4

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

   4. Applied rewrites 86.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(z\right)}} \cdot \frac{\sin y}{y} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(z\right)} \cdot \color{blue}{\frac{\sin y}{y}} \]

      6. frac-times N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \sin y}{\left(\mathsf{neg}\left(z\right)\right) \cdot y}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \sin y}{\color{blue}{\mathsf{neg}\left(z \cdot y\right)}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(x\right)\right) \cdot \sin y}{\mathsf{neg}\left(\color{blue}{z \cdot y}\right)} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\sin y}{\mathsf{neg}\left(z \cdot y\right)}} \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\sin y}{\mathsf{neg}\left(z \cdot y\right)}} \]

      11. lower-neg.f64 N/A

          \[\leadsto \color{blue}{\left(-x\right)} \cdot \frac{\sin y}{\mathsf{neg}\left(z \cdot y\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(-x\right) \cdot \color{blue}{\frac{\sin y}{\mathsf{neg}\left(z \cdot y\right)}} \]

      13. lift-*.f64 N/A

          \[\leadsto \left(-x\right) \cdot \frac{\sin y}{\mathsf{neg}\left(\color{blue}{z \cdot y}\right)} \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \left(-x\right) \cdot \frac{\sin y}{\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot y}} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(-x\right) \cdot \frac{\sin y}{\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot y}} \]

      16. lower-neg.f64 95.9

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

   6. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\left(-x\right) \cdot \frac{\sin y}{\left(-z\right) \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.005:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), y \cdot y, 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-33}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y \cdot x\_m}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= y 1.7e-33) (/ x_m z) (/ (* (sin y) x_m) (* z y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1.7e-33) {
		tmp = x_m / z;
	} else {
		tmp = (sin(y) * x_m) / (z * y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.7d-33) then
        tmp = x_m / z
    else
        tmp = (sin(y) * x_m) / (z * y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1.7e-33) {
		tmp = x_m / z;
	} else {
		tmp = (Math.sin(y) * x_m) / (z * y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= 1.7e-33:
		tmp = x_m / z
	else:
		tmp = (math.sin(y) * x_m) / (z * y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 1.7e-33)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(Float64(sin(y) * x_m) / Float64(z * y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= 1.7e-33)
		tmp = x_m / z;
	else
		tmp = (sin(y) * x_m) / (z * y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 1.7e-33], N[(x$95$m / z), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] * x$95$m), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.7e-33

   1. Initial program 95.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 68.3

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\frac{x}{z}} \]

   if 1.7e-33 < y

   1. Initial program 91.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 96.5

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

   4. Applied rewrites 96.5%

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

Alternative 8: 58.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot x\_m}{z}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+155}:\\ \;\;\;\;\frac{x\_m}{z \cdot z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{y \cdot y} \cdot x\_m\right) \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 1.22e+48)
    (/ (* (fma (* y y) -0.16666666666666666 1.0) x_m) z)
    (if (<= y 6.6e+155)
      (* (/ x_m (* z z)) z)
      (* (* (/ y (* y y)) x_m) (/ y z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1.22e+48) {
		tmp = (fma((y * y), -0.16666666666666666, 1.0) * x_m) / z;
	} else if (y <= 6.6e+155) {
		tmp = (x_m / (z * z)) * z;
	} else {
		tmp = ((y / (y * y)) * x_m) * (y / z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 1.22e+48)
		tmp = Float64(Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * x_m) / z);
	elseif (y <= 6.6e+155)
		tmp = Float64(Float64(x_m / Float64(z * z)) * z);
	else
		tmp = Float64(Float64(Float64(y / Float64(y * y)) * x_m) * Float64(y / z));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 1.22e+48], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 6.6e+155], N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(y / N[(y * y), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 1.22000000000000004e48

   1. Initial program 96.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 66.6

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

   5. Applied rewrites 66.6%

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

   if 1.22000000000000004e48 < y < 6.5999999999999997e155

   1. Initial program 97.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 23.9

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 23.9%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.9%

      \[\leadsto \frac{-1}{z} \cdot \color{blue}{\left(-x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 23.9%

      \[\leadsto \frac{\frac{-1}{z}}{\color{blue}{\frac{-1}{x}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 39.2%

       \[\leadsto \frac{-x}{-z \cdot z} \cdot \color{blue}{z} \]

   if 6.5999999999999997e155 < y

   1. Initial program 84.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 84.4

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

   4. Applied rewrites 84.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 28.4

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

   7. Applied rewrites 28.4%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{0 - x}}{\mathsf{neg}\left(y\right)} \]

      4. div-sub N/A

         \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(\frac{0}{\mathsf{neg}\left(y\right)} - \frac{x}{\mathsf{neg}\left(y\right)}\right)} \]

      5. frac-sub N/A

         \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(y\right)\right) - \left(\mathsf{neg}\left(y\right)\right) \cdot x}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{0 \cdot \left(\mathsf{neg}\left(y\right)\right) - \left(\mathsf{neg}\left(y\right)\right) \cdot x}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)}} \]

      7. lower--.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{0 \cdot \left(\mathsf{neg}\left(y\right)\right) - \left(\mathsf{neg}\left(y\right)\right) \cdot x}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{0 \cdot \left(\mathsf{neg}\left(y\right)\right)} - \left(\mathsf{neg}\left(y\right)\right) \cdot x}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)} \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{0 \cdot \color{blue}{\left(-y\right)} - \left(\mathsf{neg}\left(y\right)\right) \cdot x}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \frac{0 \cdot \left(-y\right) - \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot x}}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)} \]

      11. lower-neg.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \frac{0 \cdot \left(-y\right) - \color{blue}{\left(-y\right)} \cdot x}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \frac{0 \cdot \left(-y\right) - \left(-y\right) \cdot x}{\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(\mathsf{neg}\left(y\right)\right)}} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \frac{0 \cdot \left(-y\right) - \left(-y\right) \cdot x}{\color{blue}{\left(-y\right)} \cdot \left(\mathsf{neg}\left(y\right)\right)} \]

      14. lower-neg.f64 38.0

          \[\leadsto \frac{y}{z} \cdot \frac{0 \cdot \left(-y\right) - \left(-y\right) \cdot x}{\left(-y\right) \cdot \color{blue}{\left(-y\right)}} \]

   9. Applied rewrites 38.0%

      \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{0 \cdot \left(-y\right) - \left(-y\right) \cdot x}{\left(-y\right) \cdot \left(-y\right)}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{0 \cdot \left(-y\right) - \left(-y\right) \cdot x}{\left(-y\right) \cdot \left(-y\right)}} \]

       2. lift--.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{0 \cdot \left(-y\right) - \left(-y\right) \cdot x}}{\left(-y\right) \cdot \left(-y\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{0 \cdot \left(-y\right)} - \left(-y\right) \cdot x}{\left(-y\right) \cdot \left(-y\right)} \]

       4. mul0-lft N/A

          \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{0} - \left(-y\right) \cdot x}{\left(-y\right) \cdot \left(-y\right)} \]

       5. neg-sub0 N/A

          \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\mathsf{neg}\left(\left(-y\right) \cdot x\right)}}{\left(-y\right) \cdot \left(-y\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{neg}\left(\color{blue}{\left(-y\right) \cdot x}\right)}{\left(-y\right) \cdot \left(-y\right)} \]

       7. *-commutative N/A

          \[\leadsto \frac{y}{z} \cdot \frac{\mathsf{neg}\left(\color{blue}{x \cdot \left(-y\right)}\right)}{\left(-y\right) \cdot \left(-y\right)} \]

       8. distribute-rgt-neg-in N/A

          \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{x \cdot \left(\mathsf{neg}\left(\left(-y\right)\right)\right)}}{\left(-y\right) \cdot \left(-y\right)} \]

       9. lift-neg.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \frac{x \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)}{\left(-y\right) \cdot \left(-y\right)} \]

       10. remove-double-neg N/A

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

       11. associate-/l* N/A

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

       12. lower-*.f64 N/A

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

       13. lower-/.f64 38.4

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

       14. lift-*.f64 N/A

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

       15. lift-neg.f64 N/A

           \[\leadsto \frac{y}{z} \cdot \left(x \cdot \frac{y}{\color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \cdot \left(-y\right)}\right) \]

       16. lift-neg.f64 N/A

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

       17. sqr-neg N/A

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

       18. lower-*.f64 38.4

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

   11. Applied rewrites 38.4%

       \[\leadsto \frac{y}{z} \cdot \color{blue}{\left(x \cdot \frac{y}{y \cdot y}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot x}{z}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{y \cdot y} \cdot x\right) \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x\_m}{\frac{\left(y \cdot y\right) \cdot z}{y}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 1.22e+48)
    (/ (* (fma (* y y) -0.16666666666666666 1.0) x_m) z)
    (/ (* y x_m) (/ (* (* y y) z) y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1.22e+48) {
		tmp = (fma((y * y), -0.16666666666666666, 1.0) * x_m) / z;
	} else {
		tmp = (y * x_m) / (((y * y) * z) / y);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 1.22e+48)
		tmp = Float64(Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * x_m) / z);
	else
		tmp = Float64(Float64(y * x_m) / Float64(Float64(Float64(y * y) * z) / y));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 1.22e+48], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(y * x$95$m), $MachinePrecision] / N[(N[(N[(y * y), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.22000000000000004e48

   1. Initial program 96.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 66.6

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

   5. Applied rewrites 66.6%

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

   if 1.22000000000000004e48 < y

   1. Initial program 89.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 95.5

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 25.6

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

   7. Applied rewrites 25.6%

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

      1. remove-double-neg N/A

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

      2. lift-*.f64 N/A

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

      3. distribute-lft-neg-out N/A

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

      4. lift-neg.f64 N/A

         \[\leadsto \frac{y \cdot x}{\mathsf{neg}\left(\color{blue}{\left(-z\right)} \cdot y\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{y \cdot x}{\mathsf{neg}\left(\color{blue}{y \cdot \left(-z\right)}\right)} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(-z\right)}} \]

      7. neg-mul-1 N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{y \cdot x}{\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) \cdot \left(-z\right)} \]

      10. *-inverses N/A

          \[\leadsto \frac{y \cdot x}{\left(y \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{-y}{-y}}\right)\right)\right) \cdot \left(-z\right)} \]

      11. distribute-frac-neg2 N/A

          \[\leadsto \frac{y \cdot x}{\left(y \cdot \color{blue}{\frac{-y}{\mathsf{neg}\left(\left(-y\right)\right)}}\right) \cdot \left(-z\right)} \]

      12. lift-neg.f64 N/A

          \[\leadsto \frac{y \cdot x}{\left(y \cdot \frac{-y}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)}\right) \cdot \left(-z\right)} \]

      13. remove-double-neg N/A

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

      14. associate-/l* N/A

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

      15. remove-double-neg N/A

          \[\leadsto \frac{y \cdot x}{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(y \cdot \left(-y\right)\right)\right)\right)}}{y} \cdot \left(-z\right)} \]

      16. distribute-lft-neg-out N/A

          \[\leadsto \frac{y \cdot x}{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \left(-y\right)}\right)}{y} \cdot \left(-z\right)} \]

      17. lift-neg.f64 N/A

          \[\leadsto \frac{y \cdot x}{\frac{\mathsf{neg}\left(\color{blue}{\left(-y\right)} \cdot \left(-y\right)\right)}{y} \cdot \left(-z\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{y \cdot x}{\frac{\mathsf{neg}\left(\color{blue}{\left(-y\right) \cdot \left(-y\right)}\right)}{y} \cdot \left(-z\right)} \]

      19. associate-*l/ N/A

          \[\leadsto \frac{y \cdot x}{\color{blue}{\frac{\left(\mathsf{neg}\left(\left(-y\right) \cdot \left(-y\right)\right)\right) \cdot \left(-z\right)}{y}}} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{y \cdot x}{\color{blue}{\frac{\left(\mathsf{neg}\left(\left(-y\right) \cdot \left(-y\right)\right)\right) \cdot \left(-z\right)}{y}}} \]

   9. Applied rewrites 43.6%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\frac{\left(y \cdot y\right) \cdot z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot x\_m}{z}\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{+173}:\\ \;\;\;\;\frac{x\_m}{z \cdot z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y} \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 1.22e+48)
    (/ (* (fma (* y y) -0.16666666666666666 1.0) x_m) z)
    (if (<= y 1.16e+173) (* (/ x_m (* z z)) z) (/ (* (/ x_m y) y) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1.22e+48) {
		tmp = (fma((y * y), -0.16666666666666666, 1.0) * x_m) / z;
	} else if (y <= 1.16e+173) {
		tmp = (x_m / (z * z)) * z;
	} else {
		tmp = ((x_m / y) * y) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 1.22e+48)
		tmp = Float64(Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * x_m) / z);
	elseif (y <= 1.16e+173)
		tmp = Float64(Float64(x_m / Float64(z * z)) * z);
	else
		tmp = Float64(Float64(Float64(x_m / y) * y) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 1.22e+48], N[(N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 1.16e+173], N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(x$95$m / y), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 1.22000000000000004e48

   1. Initial program 96.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 66.6

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

   5. Applied rewrites 66.6%

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

   if 1.22000000000000004e48 < y < 1.16e173

   1. Initial program 98.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 22.0

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 22.0%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 22.0%

      \[\leadsto \frac{-1}{z} \cdot \color{blue}{\left(-x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 22.0%

      \[\leadsto \frac{\frac{-1}{z}}{\color{blue}{\frac{-1}{x}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 42.3%

       \[\leadsto \frac{-x}{-z \cdot z} \cdot \color{blue}{z} \]

   if 1.16e173 < y

   1. Initial program 83.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 9.6

         \[\leadsto \color{blue}{\frac{x}{z}} \]

   5. Applied rewrites 9.6%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
   6. Step-by-step derivation

   7. Applied rewrites 9.6%

      \[\leadsto \frac{-1}{z} \cdot \color{blue}{\left(-x\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 9.6%

      \[\leadsto \frac{\frac{1}{\frac{-1}{x}}}{\color{blue}{-z}} \]

   11. Applied rewrites 41.8%

       \[\leadsto \frac{\frac{-x}{y} \cdot y}{-\color{blue}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot x}{z}\\ \mathbf{elif}\;y \leq 1.16 \cdot 10^{+173}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m}{z} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (/ x_m z)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * (x_m / z);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * (x_m / z)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * (x_m / z);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * (x_m / z)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(x_m / z))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (x_m / z);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 54.5

      \[\leadsto \color{blue}{\frac{x}{z}} \]

5. Applied rewrites 54.5%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
6. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\sin y}\\ t_1 := \frac{x \cdot \frac{1}{t\_0}}{z}\\ \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (sin y))) (t_1 (/ (* x (/ 1.0 t_0)) z)))
   (if (< z -4.2173720203427147e-29)
     t_1
     (if (< z 4.446702369113811e+64) (/ x (* z t_0)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = y / sin(y);
	double t_1 = (x * (1.0 / t_0)) / z;
	double tmp;
	if (z < -4.2173720203427147e-29) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x / (z * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y / sin(y)
    t_1 = (x * (1.0d0 / t_0)) / z
    if (z < (-4.2173720203427147d-29)) then
        tmp = t_1
    else if (z < 4.446702369113811d+64) then
        tmp = x / (z * t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / Math.sin(y);
	double t_1 = (x * (1.0 / t_0)) / z;
	double tmp;
	if (z < -4.2173720203427147e-29) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x / (z * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / math.sin(y)
	t_1 = (x * (1.0 / t_0)) / z
	tmp = 0
	if z < -4.2173720203427147e-29:
		tmp = t_1
	elif z < 4.446702369113811e+64:
		tmp = x / (z * t_0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / sin(y))
	t_1 = Float64(Float64(x * Float64(1.0 / t_0)) / z)
	tmp = 0.0
	if (z < -4.2173720203427147e-29)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = Float64(x / Float64(z * t_0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / sin(y);
	t_1 = (x * (1.0 / t_0)) / z;
	tmp = 0.0;
	if (z < -4.2173720203427147e-29)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = x / (z * t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Less[z, -4.2173720203427147e-29], t$95$1, If[Less[z, 4.446702369113811e+64], N[(x / N[(z * t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024254 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -42173720203427147/1000000000000000000000000000000000000000000000) (/ (* x (/ 1 (/ y (sin y)))) z) (if (< z 44467023691138110000000000000000000000000000000000000000000000000) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1 (/ y (sin y)))) z))))

  (/ (* x (/ (sin y) y)) z))