Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.3% → 99.8%

Time: 8.8s

Alternatives: 12

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: 96.3% 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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{x\_m}{z} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot x\_m}{z}\\ \end{array} \end{array} \end{array} \]

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 5e-15) (* (/ 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 <= 5e-15) {
		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 <= 5d-15) 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 <= 5e-15) {
		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 <= 5e-15:
		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 <= 5e-15)
		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 <= 5e-15)
		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, 5e-15], 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 < 4.99999999999999999e-15

   1. Initial program 93.9%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 96.6

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

   4. Applied rewrites 96.6%

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

   if 4.99999999999999999e-15 < x

   1. Initial program 99.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y}{y} \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\frac{\sin y}{y} \cdot x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-165}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;t\_0 \leq 10^{-285}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x\_m}{\left(-y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

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 -1e-165)
      (* (fma (* y y) -0.16666666666666666 1.0) (/ x_m z))
      (if (<= t_0 1e-285) (* (- 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 = ((sin(y) / y) * x_m) / z;
	double tmp;
	if (t_0 <= -1e-165) {
		tmp = fma((y * y), -0.16666666666666666, 1.0) * (x_m / z);
	} else if (t_0 <= 1e-285) {
		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(Float64(sin(y) / y) * x_m) / z)
	tmp = 0.0
	if (t_0 <= -1e-165)
		tmp = Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * Float64(x_m / z));
	elseif (t_0 <= 1e-285)
		tmp = Float64(Float64(-y) * Float64(x_m / Float64(Float64(-y) * 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, -1e-165], N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-285], 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) < -1e-165

   1. Initial program 99.9%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 95.5

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

   4. Applied rewrites 95.5%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 62.4

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

   7. Applied rewrites 62.4%

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

   if -1e-165 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 1.00000000000000007e-285

   1. Initial program 86.8%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 76.6

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

   7. Applied rewrites 76.6%

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

   if 1.00000000000000007e-285 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 58.8

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

   5. Applied rewrites 58.8%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq -1 \cdot 10^{-165}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq 10^{-285}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{\left(-y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\frac{\sin y}{y} \cdot x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-165}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;t\_0 \leq 10^{-285}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x\_m}{\left(-y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

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 -1e-165)
      (* (* -0.16666666666666666 (* y y)) (/ x_m z))
      (if (<= t_0 1e-285) (* (- 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 = ((sin(y) / y) * x_m) / z;
	double tmp;
	if (t_0 <= -1e-165) {
		tmp = (-0.16666666666666666 * (y * y)) * (x_m / z);
	} else if (t_0 <= 1e-285) {
		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 = ((sin(y) / y) * x_m) / z
    if (t_0 <= (-1d-165)) then
        tmp = ((-0.16666666666666666d0) * (y * y)) * (x_m / z)
    else if (t_0 <= 1d-285) 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 = ((Math.sin(y) / y) * x_m) / z;
	double tmp;
	if (t_0 <= -1e-165) {
		tmp = (-0.16666666666666666 * (y * y)) * (x_m / z);
	} else if (t_0 <= 1e-285) {
		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 = ((math.sin(y) / y) * x_m) / z
	tmp = 0
	if t_0 <= -1e-165:
		tmp = (-0.16666666666666666 * (y * y)) * (x_m / z)
	elif t_0 <= 1e-285:
		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(Float64(sin(y) / y) * x_m) / z)
	tmp = 0.0
	if (t_0 <= -1e-165)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(y * y)) * Float64(x_m / z));
	elseif (t_0 <= 1e-285)
		tmp = Float64(Float64(-y) * Float64(x_m / Float64(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 = ((sin(y) / y) * x_m) / z;
	tmp = 0.0;
	if (t_0 <= -1e-165)
		tmp = (-0.16666666666666666 * (y * y)) * (x_m / z);
	elseif (t_0 <= 1e-285)
		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[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -1e-165], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-285], 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) < -1e-165

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}}{z} \]
   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 62.4

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

   5. Applied rewrites 62.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-/r/ N/A

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

      7. clear-num N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 62.4

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

   7. Applied rewrites 62.4%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 8.1%

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

   if -1e-165 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 1.00000000000000007e-285

   1. Initial program 86.8%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 76.6

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

   7. Applied rewrites 76.6%

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

   if 1.00000000000000007e-285 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 58.8

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

   5. Applied rewrites 58.8%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq -1 \cdot 10^{-165}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq 10^{-285}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{\left(-y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_0 \cdot x\_m}{z} \leq -200000000:\\ \;\;\;\;\frac{x\_m}{z \cdot y} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot t\_0\\ \end{array} \end{array} \end{array} \]

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) -200000000.0)
      (* (/ x_m (* z y)) (sin y))
      (* (/ 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) <= -200000000.0) {
		tmp = (x_m / (z * y)) * sin(y);
	} 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) <= (-200000000.0d0)) then
        tmp = (x_m / (z * y)) * sin(y)
    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) <= -200000000.0) {
		tmp = (x_m / (z * y)) * Math.sin(y);
	} 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) <= -200000000.0:
		tmp = (x_m / (z * y)) * math.sin(y)
	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) <= -200000000.0)
		tmp = Float64(Float64(x_m / Float64(z * y)) * sin(y));
	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) <= -200000000.0)
		tmp = (x_m / (z * y)) * sin(y);
	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], -200000000.0], N[(N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $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) < -2e8

   1. Initial program 99.9%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 76.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-neg.f64 N/A

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

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

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. times-frac N/A

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

      12. lift-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. metadata-eval N/A

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

      17. *-rgt-identity N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   6. Applied rewrites 76.1%

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

   if -2e8 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 93.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 97.0

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

   4. Applied rewrites 97.0%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq -200000000:\\ \;\;\;\;\frac{x}{z \cdot y} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x\_m}{z} \leq -2 \cdot 10^{-66}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot z, 0.16666666666666666, z\right)}\\ \end{array} \end{array} \]

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) -2e-66)
    (/ x_m (/ z (fma (* y y) -0.16666666666666666 1.0)))
    (/ x_m (fma (* (* y y) z) 0.16666666666666666 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) <= -2e-66) {
		tmp = x_m / (z / fma((y * y), -0.16666666666666666, 1.0));
	} else {
		tmp = x_m / fma(((y * y) * z), 0.16666666666666666, 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) <= -2e-66)
		tmp = Float64(x_m / Float64(z / fma(Float64(y * y), -0.16666666666666666, 1.0)));
	else
		tmp = Float64(x_m / fma(Float64(Float64(y * y) * z), 0.16666666666666666, 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[N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], -2e-66], N[(x$95$m / N[(z / N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(N[(y * y), $MachinePrecision] * z), $MachinePrecision] * 0.16666666666666666 + z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -2e-66

   1. Initial program 99.9%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 60.6

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

   5. Applied rewrites 60.6%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)}}{z} \]
   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. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 60.6

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

   7. Applied rewrites 60.6%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.5

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

   10. Applied rewrites 63.5%

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

   if -2e-66 < (/.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. 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 46.7

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

   5. Applied rewrites 46.7%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)}}{z} \]
   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. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 46.7

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

   7. Applied rewrites 46.7%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 68.2

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

   10. Applied rewrites 68.2%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq -2 \cdot 10^{-66}:\\ \;\;\;\;\frac{x}{\frac{z}{\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot z, 0.16666666666666666, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x\_m}{z} \leq -2 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot z, 0.16666666666666666, z\right)}\\ \end{array} \end{array} \]

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) -2e-66)
    (* (fma (* y y) -0.16666666666666666 1.0) (/ x_m z))
    (/ x_m (fma (* (* y y) z) 0.16666666666666666 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) <= -2e-66) {
		tmp = fma((y * y), -0.16666666666666666, 1.0) * (x_m / z);
	} else {
		tmp = x_m / fma(((y * y) * z), 0.16666666666666666, 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) <= -2e-66)
		tmp = Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * Float64(x_m / z));
	else
		tmp = Float64(x_m / fma(Float64(Float64(y * y) * z), 0.16666666666666666, 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[N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], -2e-66], N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(N[(y * y), $MachinePrecision] * z), $MachinePrecision] * 0.16666666666666666 + z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -2e-66

   1. Initial program 99.9%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 94.6

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

   4. Applied rewrites 94.6%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 63.5

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

   7. Applied rewrites 63.5%

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

   if -2e-66 < (/.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. 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 46.7

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

   5. Applied rewrites 46.7%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)}}{z} \]
   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. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 46.7

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

   7. Applied rewrites 46.7%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 68.2

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

   10. Applied rewrites 68.2%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq -2 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot z, 0.16666666666666666, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x\_m}{z} \leq 5 \cdot 10^{-310}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{z \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

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-310)
    (/ (* x_m x_m) (* z x_m))
    (/ 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-310) {
		tmp = (x_m * x_m) / (z * x_m);
	} 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-310) then
        tmp = (x_m * x_m) / (z * x_m)
    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-310) {
		tmp = (x_m * x_m) / (z * x_m);
	} 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-310:
		tmp = (x_m * x_m) / (z * x_m)
	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-310)
		tmp = Float64(Float64(x_m * x_m) / Float64(z * x_m));
	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-310)
		tmp = (x_m * x_m) / (z * x_m);
	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-310], N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(z * x$95$m), $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) < 4.999999999999985e-310

   1. Initial program 93.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 56.0

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

   5. Applied rewrites 56.0%

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

   7. Applied rewrites 56.5%

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

   9. Applied rewrites 56.4%

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

   11. Applied rewrites 46.2%

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

   if 4.999999999999985e-310 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 58.0

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

   5. Applied rewrites 58.0%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq 5 \cdot 10^{-310}:\\ \;\;\;\;\frac{x \cdot x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x\_m}{z} \leq 5 \cdot 10^{-310}:\\ \;\;\;\;\frac{y \cdot x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

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) 5e-310)
    (/ (* 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) <= 5e-310) {
		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) <= 5d-310) 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) <= 5e-310) {
		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) <= 5e-310:
		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) <= 5e-310)
		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) <= 5e-310)
		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], 5e-310], 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) < 4.999999999999985e-310

   1. Initial program 93.2%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 87.2

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

   4. Applied rewrites 87.2%

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

   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 50.5

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

   7. Applied rewrites 50.5%

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

   if 4.999999999999985e-310 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 58.0

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

   5. Applied rewrites 58.0%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq 5 \cdot 10^{-310}:\\ \;\;\;\;\frac{y \cdot x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 5 \cdot 10^{-92}:\\ \;\;\;\;\frac{x\_m}{z \cdot z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

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) 5e-92) (* (/ 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) <= 5e-92) {
		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) <= 5d-92) 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) <= 5e-92) {
		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) <= 5e-92:
		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(sin(y) / y) <= 5e-92)
		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) <= 5e-92)
		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[Sin[y], $MachinePrecision] / y), $MachinePrecision], 5e-92], 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 (sin.f64 y) y) < 5.00000000000000011e-92

   1. Initial program 90.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 15.1

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

   5. Applied rewrites 15.1%

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

   7. Applied rewrites 16.2%

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

   9. Applied rewrites 16.2%

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

   11. Applied rewrites 21.7%

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

   if 5.00000000000000011e-92 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 93.2

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

   5. Applied rewrites 93.2%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 5 \cdot 10^{-92}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 0.00024:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot x\_m, -0.16666666666666666, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z \cdot y} \cdot \sin y\\ \end{array} \end{array} \]

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 0.00024)
    (/ (fma (* (* y y) x_m) -0.16666666666666666 x_m) z)
    (* (/ x_m (* z y)) (sin 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 <= 0.00024) {
		tmp = fma(((y * y) * x_m), -0.16666666666666666, x_m) / z;
	} else {
		tmp = (x_m / (z * y)) * sin(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 <= 0.00024)
		tmp = Float64(fma(Float64(Float64(y * y) * x_m), -0.16666666666666666, x_m) / z);
	else
		tmp = Float64(Float64(x_m / Float64(z * y)) * sin(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, 0.00024], N[(N[(N[(N[(y * y), $MachinePrecision] * x$95$m), $MachinePrecision] * -0.16666666666666666 + x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.40000000000000006e-4

   1. Initial program 97.4%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. 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 83.6

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

   4. Applied rewrites 83.6%

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

   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 59.3

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

   7. Applied rewrites 59.3%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. lft-mult-inverse N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. associate-*r/ N/A

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

   10. Applied rewrites 69.9%

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

   if 2.40000000000000006e-4 < y

   1. Initial program 88.4%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-neg.f64 N/A

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

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

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. times-frac N/A

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

      12. lift-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. metadata-eval N/A

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

      17. *-rgt-identity N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

   6. Applied rewrites 94.3%

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

Alternative 11: 62.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x\_m}{\left(-y\right) \cdot z}\\ \end{array} \end{array} \]

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 7.8e-5) (/ x_m z) (* (- 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 <= 7.8e-5) {
		tmp = x_m / z;
	} else {
		tmp = -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 <= 7.8d-5) then
        tmp = x_m / z
    else
        tmp = -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 <= 7.8e-5) {
		tmp = x_m / z;
	} else {
		tmp = -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 <= 7.8e-5:
		tmp = x_m / z
	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 <= 7.8e-5)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(Float64(-y) * Float64(x_m / Float64(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 <= 7.8e-5)
		tmp = x_m / z;
	else
		tmp = -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, 7.8e-5], N[(x$95$m / z), $MachinePrecision], N[((-y) * N[(x$95$m / N[((-y) * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 7.7999999999999999e-5

   1. Initial program 97.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if 7.7999999999999999e-5 < y

   1. Initial program 88.4%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 94.3%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 31.0

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

   7. Applied rewrites 31.0%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{\left(-y\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.1% 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 95.0%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 56.6

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

5. Applied rewrites 56.6%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
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 2024270 
(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))