Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 95.9% → 99.6%

Time: 11.7s

Alternatives: 10

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: 95.9% 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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.2 \cdot 10^{-61}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot t\_0}{z}\\ \end{array} \end{array} \end{array} \]

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 3.2e-61) (* t_0 (/ x_m z)) (/ (* x_m t_0) 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 <= 3.2e-61) {
		tmp = t_0 * (x_m / z);
	} else {
		tmp = (x_m * t_0) / 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 <= 3.2d-61) then
        tmp = t_0 * (x_m / z)
    else
        tmp = (x_m * t_0) / 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 <= 3.2e-61) {
		tmp = t_0 * (x_m / z);
	} else {
		tmp = (x_m * t_0) / 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 <= 3.2e-61:
		tmp = t_0 * (x_m / z)
	else:
		tmp = (x_m * t_0) / 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 <= 3.2e-61)
		tmp = Float64(t_0 * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m * t_0) / 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 <= 3.2e-61)
		tmp = t_0 * (x_m / z);
	else
		tmp = (x_m * t_0) / 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, 3.2e-61], N[(t$95$0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2000000000000001e-61

   1. Initial program 93.8%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 98.3

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

   4. Applied rewrites 98.3%

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

   if 3.2000000000000001e-61 < 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. Add Preprocessing

Alternative 2: 49.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-128}:\\ \;\;\;\;x\_m \cdot \frac{-0.16666666666666666 \cdot \left(y \cdot y\right)}{z}\\ \mathbf{elif}\;t\_0 \leq 10^{-308}:\\ \;\;\;\;y \cdot \frac{x\_m}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

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 (/ (* x_m (/ (sin y) y)) z)))
   (*
    x_s
    (if (<= t_0 -2e-128)
      (* x_m (/ (* -0.16666666666666666 (* y y)) z))
      (if (<= t_0 1e-308) (* y (/ x_m (* y 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 = (x_m * (sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -2e-128) {
		tmp = x_m * ((-0.16666666666666666 * (y * y)) / z);
	} else if (t_0 <= 1e-308) {
		tmp = y * (x_m / (y * 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) :: t_0
    real(8) :: tmp
    t_0 = (x_m * (sin(y) / y)) / z
    if (t_0 <= (-2d-128)) then
        tmp = x_m * (((-0.16666666666666666d0) * (y * y)) / z)
    else if (t_0 <= 1d-308) then
        tmp = y * (x_m / (y * 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 t_0 = (x_m * (Math.sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -2e-128) {
		tmp = x_m * ((-0.16666666666666666 * (y * y)) / z);
	} else if (t_0 <= 1e-308) {
		tmp = y * (x_m / (y * 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):
	t_0 = (x_m * (math.sin(y) / y)) / z
	tmp = 0
	if t_0 <= -2e-128:
		tmp = x_m * ((-0.16666666666666666 * (y * y)) / z)
	elif t_0 <= 1e-308:
		tmp = y * (x_m / (y * 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(x_m * Float64(sin(y) / y)) / z)
	tmp = 0.0
	if (t_0 <= -2e-128)
		tmp = Float64(x_m * Float64(Float64(-0.16666666666666666 * Float64(y * y)) / z));
	elseif (t_0 <= 1e-308)
		tmp = Float64(y * Float64(x_m / Float64(y * 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)
	t_0 = (x_m * (sin(y) / y)) / z;
	tmp = 0.0;
	if (t_0 <= -2e-128)
		tmp = x_m * ((-0.16666666666666666 * (y * y)) / z);
	elseif (t_0 <= 1e-308)
		tmp = y * (x_m / (y * 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_] := Block[{t$95$0 = N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -2e-128], N[(x$95$m * N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-308], N[(y * N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision]), $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) < -2.00000000000000011e-128

   1. Initial program 99.7%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 85.4

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

   4. Applied rewrites 85.4%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 58.6

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

   7. Applied rewrites 58.6%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 5.5

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

   10. Applied rewrites 5.5%

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

   if -2.00000000000000011e-128 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 9.9999999999999991e-309

   1. Initial program 89.5%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-sin.f64 N/A

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

      2. frac-times N/A

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

      3. lift-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 91.3

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

   6. Applied rewrites 91.3%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 54.6

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

   9. Applied rewrites 54.6%

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

       1. lift-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 72.3

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

   11. Applied rewrites 72.3%

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

   if 9.9999999999999991e-309 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 56.3

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

   5. Applied rewrites 56.3%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -2 \cdot 10^{-128}:\\ \;\;\;\;x \cdot \frac{-0.16666666666666666 \cdot \left(y \cdot y\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 10^{-308}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot t\_0}{z} \leq -2 \cdot 10^{+38}:\\ \;\;\;\;\sin y \cdot \frac{x\_m}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

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_s
    (if (<= (/ (* x_m t_0) z) -2e+38)
      (* (sin y) (/ x_m (* y z)))
      (* 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 * t_0) / z) <= -2e+38) {
		tmp = sin(y) * (x_m / (y * z));
	} 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 * t_0) / z) <= (-2d+38)) then
        tmp = sin(y) * (x_m / (y * z))
    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 * t_0) / z) <= -2e+38) {
		tmp = Math.sin(y) * (x_m / (y * z));
	} 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 * t_0) / z) <= -2e+38:
		tmp = math.sin(y) * (x_m / (y * z))
	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 (Float64(Float64(x_m * t_0) / z) <= -2e+38)
		tmp = Float64(sin(y) * Float64(x_m / Float64(y * z)));
	else
		tmp = Float64(t_0 * 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)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (((x_m * t_0) / z) <= -2e+38)
		tmp = sin(y) * (x_m / (y * z));
	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[N[(N[(x$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision], -2e+38], N[(N[Sin[y], $MachinePrecision] * N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -1.99999999999999995e38

   1. Initial program 99.7%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. *-lft-identity N/A

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

      7. associate-*l/ N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-lft-identity N/A

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

      14. associate-/l/ N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 83.0

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

   4. Applied rewrites 83.0%

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

   if -1.99999999999999995e38 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 94.7%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 96.7

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

   4. Applied rewrites 96.7%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -2 \cdot 10^{+38}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{z \cdot \frac{y}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(y, y \cdot -0.16666666666666666, -1\right) \cdot \frac{z}{x\_m}}\\ \end{array} \end{array} \]

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 (<= (/ (* x_m (/ (sin y) y)) z) -1e-150)
    (/ (fma -0.16666666666666666 (* y (* y y)) y) (* z (/ y x_m)))
    (/ -1.0 (* (fma y (* y -0.16666666666666666) -1.0) (/ z 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 (((x_m * (sin(y) / y)) / z) <= -1e-150) {
		tmp = fma(-0.16666666666666666, (y * (y * y)), y) / (z * (y / x_m));
	} else {
		tmp = -1.0 / (fma(y, (y * -0.16666666666666666), -1.0) * (z / 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 (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= -1e-150)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * Float64(y * y)), y) / Float64(z * Float64(y / x_m)));
	else
		tmp = Float64(-1.0 / Float64(fma(y, Float64(y * -0.16666666666666666), -1.0) * Float64(z / 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[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], -1e-150], N[(N[(-0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / N[(z * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision] * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -1.00000000000000001e-150

   1. Initial program 99.7%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-/r/ N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/l/ N/A

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

      12. remove-double-div N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 N/A

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

      15. div-inv N/A

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

      16. remove-double-div N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 83.4

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

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 50.0

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

   7. Applied rewrites 50.0%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 42.0

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

   9. Applied rewrites 42.0%

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

   if -1.00000000000000001e-150 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 94.0%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 91.0

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 46.8

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

   7. Applied rewrites 46.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. un-div-inv N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-fma.f64 N/A

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

      7. flip-+ N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

   9. Applied rewrites 45.4%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 66.6%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{z \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(y, y \cdot -0.16666666666666666, -1\right) \cdot \frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq -1 \cdot 10^{-58}:\\ \;\;\;\;x\_m \cdot \frac{-0.16666666666666666 \cdot \left(y \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(y, y \cdot -0.16666666666666666, -1\right) \cdot \frac{z}{x\_m}}\\ \end{array} \end{array} \]

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 (<= (/ (* x_m (/ (sin y) y)) z) -1e-58)
    (* x_m (/ (* -0.16666666666666666 (* y y)) z))
    (/ -1.0 (* (fma y (* y -0.16666666666666666) -1.0) (/ z 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 (((x_m * (sin(y) / y)) / z) <= -1e-58) {
		tmp = x_m * ((-0.16666666666666666 * (y * y)) / z);
	} else {
		tmp = -1.0 / (fma(y, (y * -0.16666666666666666), -1.0) * (z / 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 (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= -1e-58)
		tmp = Float64(x_m * Float64(Float64(-0.16666666666666666 * Float64(y * y)) / z));
	else
		tmp = Float64(-1.0 / Float64(fma(y, Float64(y * -0.16666666666666666), -1.0) * Float64(z / 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[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], -1e-58], N[(x$95$m * N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision] * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -1e-58

   1. Initial program 99.7%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 89.1

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

   4. Applied rewrites 89.1%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 57.6

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

   7. Applied rewrites 57.6%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 6.0

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

   10. Applied rewrites 6.0%

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

   if -1e-58 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 94.4%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 89.3

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

   4. Applied rewrites 89.3%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 47.9

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

   7. Applied rewrites 47.9%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. un-div-inv N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. lift-fma.f64 N/A

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

      7. flip-+ N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

   9. Applied rewrites 46.6%

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

   10. Taylor expanded in y around 0

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

   12. Applied rewrites 66.1%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -1 \cdot 10^{-58}:\\ \;\;\;\;x \cdot \frac{-0.16666666666666666 \cdot \left(y \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(y, y \cdot -0.16666666666666666, -1\right) \cdot \frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-109}:\\ \;\;\;\;y \cdot \frac{x\_m}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

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) 1e-109) (* y (/ x_m (* y 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) <= 1e-109) {
		tmp = y * (x_m / (y * 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) <= 1d-109) then
        tmp = y * (x_m / (y * 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) <= 1e-109) {
		tmp = y * (x_m / (y * 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) <= 1e-109:
		tmp = y * (x_m / (y * 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(sin(y) / y) <= 1e-109)
		tmp = Float64(y * Float64(x_m / Float64(y * 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) <= 1e-109)
		tmp = y * (x_m / (y * 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[Sin[y], $MachinePrecision] / y), $MachinePrecision], 1e-109], N[(y * N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 9.9999999999999999e-110

   1. Initial program 90.9%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 89.7

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

   4. Applied rewrites 89.7%

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

      1. lift-sin.f64 N/A

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

      2. frac-times N/A

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

      3. lift-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 89.7

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

   6. Applied rewrites 89.7%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 20.9

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

   9. Applied rewrites 20.9%

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

       1. lift-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 32.3

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

   11. Applied rewrites 32.3%

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

   if 9.9999999999999999e-110 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 86.8

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

   5. Applied rewrites 86.8%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-109}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x\_m}{y \cdot z}\\ \end{array} \end{array} \]

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 3.7e-8) (/ x_m z) (* (sin 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 <= 3.7e-8) {
		tmp = x_m / z;
	} else {
		tmp = sin(y) * (x_m / (y * 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 (y <= 3.7d-8) then
        tmp = x_m / z
    else
        tmp = sin(y) * (x_m / (y * 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 (y <= 3.7e-8) {
		tmp = x_m / z;
	} else {
		tmp = Math.sin(y) * (x_m / (y * 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 y <= 3.7e-8:
		tmp = x_m / z
	else:
		tmp = math.sin(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 <= 3.7e-8)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(sin(y) * Float64(x_m / Float64(y * 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 (y <= 3.7e-8)
		tmp = x_m / z;
	else
		tmp = sin(y) * (x_m / (y * 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[y, 3.7e-8], N[(x$95$m / z), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.7e-8

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if 3.7e-8 < y

   1. Initial program 88.8%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. *-lft-identity N/A

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

      7. associate-*l/ N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-lft-identity N/A

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

      14. associate-/l/ N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 92.7

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

   4. Applied rewrites 92.7%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 10^{+73}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y \cdot \frac{z}{x\_m}}\\ \end{array} \end{array} \]

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 1e+73)
    (* (/ x_m z) (fma -0.16666666666666666 (* y y) 1.0))
    (/ y (* y (/ z 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 <= 1e+73) {
		tmp = (x_m / z) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = y / (y * (z / 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 <= 1e+73)
		tmp = Float64(Float64(x_m / z) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(y / Float64(y * Float64(z / 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, 1e+73], N[(N[(x$95$m / z), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y / N[(y * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999983e72

   1. Initial program 97.3%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 65.1

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

   7. Applied rewrites 65.1%

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

   if 9.99999999999999983e72 < y

   1. Initial program 89.0%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 85.2

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

   4. Applied rewrites 85.2%

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

      1. lift-sin.f64 N/A

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

      2. frac-times N/A

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

      3. lift-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 90.7

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

   6. Applied rewrites 90.7%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 21.6

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

   9. Applied rewrites 21.6%

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

       1. lift-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. clear-num N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*l/ N/A

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

       6. lift-/.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. un-div-inv N/A

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

       9. lower-/.f64 34.6

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

       10. lift-*.f64 N/A

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

       11. lift-/.f64 N/A

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

       12. associate-*l/ N/A

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

       13. associate-/l* N/A

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

       14. lift-/.f64 N/A

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

       15. lower-*.f64 34.0

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

   11. Applied rewrites 34.0%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+73}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y \cdot \frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 10^{+73}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{y \cdot z}\\ \end{array} \end{array} \]

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 1e+73)
    (* (/ x_m z) (fma -0.16666666666666666 (* y y) 1.0))
    (* 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 <= 1e+73) {
		tmp = (x_m / z) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = 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 <= 1e+73)
		tmp = Float64(Float64(x_m / z) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(y * Float64(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, 1e+73], N[(N[(x$95$m / z), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999983e72

   1. Initial program 97.3%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 65.1

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

   7. Applied rewrites 65.1%

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

   if 9.99999999999999983e72 < y

   1. Initial program 89.0%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 85.2

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

   4. Applied rewrites 85.2%

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

      1. lift-sin.f64 N/A

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

      2. frac-times N/A

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

      3. lift-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 90.7

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

   6. Applied rewrites 90.7%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 21.6

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

   9. Applied rewrites 21.6%

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

       1. lift-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 31.5

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

   11. Applied rewrites 31.5%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+73}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.6% 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 95.6%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 55.7

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

5. Applied rewrites 55.7%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
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 2024216 
(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))