Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.2% → 99.7%

Time: 11.4s

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: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.05 \cdot 10^{+33}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{\frac{y}{\sin y}}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.05e+33)
    (* (/ (sin y) y) (/ x_m z))
    (/ (/ x_m (/ y (sin 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 (x_m <= 1.05e+33) {
		tmp = (sin(y) / y) * (x_m / z);
	} else {
		tmp = (x_m / (y / sin(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 (x_m <= 1.05d+33) then
        tmp = (sin(y) / y) * (x_m / z)
    else
        tmp = (x_m / (y / sin(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 (x_m <= 1.05e+33) {
		tmp = (Math.sin(y) / y) * (x_m / z);
	} else {
		tmp = (x_m / (y / Math.sin(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 x_m <= 1.05e+33:
		tmp = (math.sin(y) / y) * (x_m / z)
	else:
		tmp = (x_m / (y / math.sin(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 (x_m <= 1.05e+33)
		tmp = Float64(Float64(sin(y) / y) * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m / Float64(y / sin(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 (x_m <= 1.05e+33)
		tmp = (sin(y) / y) * (x_m / z);
	else
		tmp = (x_m / (y / sin(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[x$95$m, 1.05e+33], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.05e33

   1. Initial program 95.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin y}{y}\right), \color{blue}{\left(\frac{x}{z}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin y, y\right), \left(\frac{\color{blue}{x}}{z}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right), \left(\frac{x}{z}\right)\right) \]

      6. /-lowering-/.f64 97.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right), \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 97.9%

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

   if 1.05e33 < x

   1. Initial program 99.6%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{\frac{y}{\sin y}}\right), z\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{\frac{y}{\sin y}}\right), z\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y}{\sin y}\right)\right), z\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \sin y\right)\right), z\right) \]

      5. sin-lowering-sin.f64 99.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{sin.f64}\left(y\right)\right)\right), z\right) \]

   4. Applied egg-rr 99.7%

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

Alternative 2: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

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 (<= t_0 4e-163) (* (/ (sin y) z) (/ x_m y)) (* 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 (t_0 <= 4e-163) {
		tmp = (sin(y) / z) * (x_m / y);
	} 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 (t_0 <= 4d-163) then
        tmp = (sin(y) / z) * (x_m / y)
    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 (t_0 <= 4e-163) {
		tmp = (Math.sin(y) / z) * (x_m / y);
	} 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 t_0 <= 4e-163:
		tmp = (math.sin(y) / z) * (x_m / y)
	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 (t_0 <= 4e-163)
		tmp = Float64(Float64(sin(y) / z) * Float64(x_m / y));
	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 (t_0 <= 4e-163)
		tmp = (sin(y) / z) * (x_m / y);
	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[t$95$0, 4e-163], N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 3.99999999999999969e-163

   1. Initial program 97.2%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*l/ N/A

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin y}{z}\right), \color{blue}{\left(\frac{x}{y}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin y, z\right), \left(\frac{\color{blue}{x}}{y}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), z\right), \left(\frac{x}{y}\right)\right) \]

      8. /-lowering-/.f64 97.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), z\right), \mathsf{/.f64}\left(x, \color{blue}{y}\right)\right) \]

   4. Applied egg-rr 97.2%

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

   if 3.99999999999999969e-163 < (/.f64 (sin.f64 y) y)

   1. Initial program 96.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin y}{y}\right), \color{blue}{\left(\frac{x}{z}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin y, y\right), \left(\frac{\color{blue}{x}}{z}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right), \left(\frac{x}{z}\right)\right) \]

      6. /-lowering-/.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right), \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 99.4%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-57}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot t\_0}{z}\\ \end{array} \end{array} \end{array} \]

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 5e-57) (* 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 <= 5e-57) {
		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 <= 5d-57) 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 <= 5e-57) {
		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 <= 5e-57:
		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 <= 5e-57)
		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 <= 5e-57)
		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, 5e-57], 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 < 5.0000000000000002e-57

   1. Initial program 95.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin y}{y}\right), \color{blue}{\left(\frac{x}{z}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin y, y\right), \left(\frac{\color{blue}{x}}{z}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right), \left(\frac{x}{z}\right)\right) \]

      6. /-lowering-/.f64 97.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right), \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 97.6%

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

   if 5.0000000000000002e-57 < 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 4: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-52}:\\ \;\;\;\;x\_m \cdot \frac{t\_0}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

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 (<= z 4e-52) (* x_m (/ t_0 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 (z <= 4e-52) {
		tmp = x_m * (t_0 / 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 (z <= 4d-52) then
        tmp = x_m * (t_0 / 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 (z <= 4e-52) {
		tmp = x_m * (t_0 / 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 z <= 4e-52:
		tmp = x_m * (t_0 / 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 (z <= 4e-52)
		tmp = Float64(x_m * Float64(t_0 / 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 (z <= 4e-52)
		tmp = x_m * (t_0 / 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[z, 4e-52], N[(x$95$m * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4e-52

   1. Initial program 95.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\sin y}{y}}{z}\right), \color{blue}{x}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\sin y}{y}\right), z\right), x\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\sin y, y\right), z\right), x\right) \]

      6. sin-lowering-sin.f64 98.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right), z\right), x\right) \]

   4. Applied egg-rr 98.3%

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

   if 4e-52 < z

   1. Initial program 99.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin y}{y}\right), \color{blue}{\left(\frac{x}{z}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin y, y\right), \left(\frac{\color{blue}{x}}{z}\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right), \left(\frac{x}{z}\right)\right) \]

      6. /-lowering-/.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right), \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 99.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-52}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.9% accurate, 1.0× speedup

Math

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

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 (* (/ (sin y) 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) {
	return x_s * ((sin(y) / y) * (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 * ((sin(y) / y) * (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 * ((Math.sin(y) / y) * (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 * ((math.sin(y) / y) * (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(Float64(sin(y) / y) * 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 * ((sin(y) / y) * (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[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.7%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin y}{y}\right), \color{blue}{\left(\frac{x}{z}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin y, y\right), \left(\frac{\color{blue}{x}}{z}\right)\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right), \left(\frac{x}{z}\right)\right) \]

   6. /-lowering-/.f64 95.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right), \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right) \]

4. Applied egg-rr 95.5%

   \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \frac{x}{z}} \]
5. Add Preprocessing

Alternative 6: 66.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{\frac{x\_m}{z}}{\left(0 - -1\right) - -0.16666666666666666 \cdot \left(y \cdot y\right)} \end{array} \]

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) (- (- 0.0 -1.0) (* -0.16666666666666666 (* y y))))))

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) / ((0.0 - -1.0) - (-0.16666666666666666 * (y * y))));
}

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) / ((0.0d0 - (-1.0d0)) - ((-0.16666666666666666d0) * (y * y))))
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) / ((0.0 - -1.0) - (-0.16666666666666666 * (y * y))));
}

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) / ((0.0 - -1.0) - (-0.16666666666666666 * (y * y))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(Float64(x_m / z) / Float64(Float64(0.0 - -1.0) - Float64(-0.16666666666666666 * Float64(y * y)))))
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) / ((0.0 - -1.0) - (-0.16666666666666666 * (y * y))));
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[(N[(x$95$m / z), $MachinePrecision] / N[(N[(0.0 - -1.0), $MachinePrecision] - N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.7%

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

   1. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{\frac{y}{\sin y}}\right), z\right) \]

   2. un-div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{\frac{y}{\sin y}}\right), z\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y}{\sin y}\right)\right), z\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \sin y\right)\right), z\right) \]

   5. sin-lowering-sin.f64 96.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{sin.f64}\left(y\right)\right)\right), z\right) \]

4. Applied egg-rr 96.7%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)}\right), z\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right), z\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right), z\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right)\right), z\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right)\right), z\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right)\right), z\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), z\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \frac{7}{360}\right)\right)\right)\right)\right)\right), z\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{7}{360}\right)\right)\right)\right)\right)\right), z\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{7}{360}\right)\right)\right)\right)\right)\right), z\right) \]

   10. *-lowering-*.f64 66.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{7}{360}\right)\right)\right)\right)\right)\right), z\right) \]

7. Simplified 66.1%

   \[\leadsto \frac{\frac{x}{\color{blue}{1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.019444444444444445\right)\right)}}}{z} \]

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right), z\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right)\right), z\right) \]

   2. *-lowering-*.f64 66.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right)\right), z\right) \]

10. Simplified 66.6%

    \[\leadsto \frac{\frac{x}{1 + y \cdot \color{blue}{\left(y \cdot 0.16666666666666666\right)}}}{z} \]

12. Applied egg-rr 67.3%

    \[\leadsto \color{blue}{\frac{\frac{x}{0 - z}}{-0.16666666666666666 \cdot \left(y \cdot y\right) + -1}} \]

13. Final simplification 67.3%

    \[\leadsto \frac{\frac{x}{z}}{\left(0 - -1\right) - -0.16666666666666666 \cdot \left(y \cdot y\right)} \]
14. Add Preprocessing

Alternative 7: 62.1% accurate, 8.9× speedup

Math

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

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-5) (/ x_m z) (/ 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-5) {
		tmp = x_m / z;
	} else {
		tmp = y / (y * (z / x_m));
	}
	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 <= 1d-5) then
        tmp = x_m / z
    else
        tmp = y / (y * (z / x_m))
    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 <= 1e-5) {
		tmp = x_m / z;
	} else {
		tmp = y / (y * (z / x_m));
	}
	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 <= 1e-5:
		tmp = x_m / z
	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-5)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(y / Float64(y * Float64(z / x_m)));
	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 <= 1e-5)
		tmp = x_m / z;
	else
		tmp = y / (y * (z / x_m));
	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, 1e-5], N[(x$95$m / z), $MachinePrecision], N[(y / N[(y * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000008e-5

   1. Initial program 97.5%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 77.2%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 77.2%

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

   if 1.00000000000000008e-5 < y

   1. Initial program 94.5%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 17.4%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 17.4%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{z}{x}\right)}\right) \]

      3. /-lowering-/.f64 17.5%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

   7. Applied egg-rr 17.5%

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

      1. div-inv N/A

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

      2. *-inverses N/A

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

      3. frac-times N/A

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

      4. *-rgt-identity N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(y \cdot \frac{z}{x}\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{z}{x}\right)}\right)\right) \]

      7. /-lowering-/.f64 32.8%

         \[\leadsto \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right)\right) \]

   9. Applied egg-rr 32.8%

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

Alternative 8: 62.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{y \cdot z}\\ \end{array} \end{array} \]

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 1.05e-8) (/ 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 <= 1.05e-8) {
		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 <= 1.05d-8) 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 <= 1.05e-8) {
		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 <= 1.05e-8:
		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 <= 1.05e-8)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(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 <= 1.05e-8)
		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, 1.05e-8], 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 < 1.04999999999999997e-8

   1. Initial program 97.5%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 77.1%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 77.1%

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

   if 1.04999999999999997e-8 < y

   1. Initial program 94.6%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 18.6%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 18.6%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{z}{x}\right)}\right) \]

      3. /-lowering-/.f64 18.6%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

   7. Applied egg-rr 18.6%

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

      1. div-inv N/A

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

      2. *-inverses N/A

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

      3. clear-num N/A

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

      4. clear-num N/A

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

      5. times-frac N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. /-rgt-identity N/A

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

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{x}{z}}{y}\right), \color{blue}{y}\right) \]

      10. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z} \cdot \frac{1}{y}\right), y\right) \]

      11. frac-times N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{x \cdot 1}{z \cdot y}\right), y\right) \]

      12. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{z \cdot y}\right), y\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(z \cdot y\right)\right), y\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y \cdot z\right)\right), y\right) \]

      15. *-lowering-*.f64 32.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), y\right) \]

   9. Applied egg-rr 32.6%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{\frac{x\_m}{1 + y \cdot \left(y \cdot 0.16666666666666666\right)}}{z} \end{array} \]

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 (+ 1.0 (* y (* y 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) {
	return x_s * ((x_m / (1.0 + (y * (y * 0.16666666666666666)))) / 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 / (1.0d0 + (y * (y * 0.16666666666666666d0)))) / 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 / (1.0 + (y * (y * 0.16666666666666666)))) / 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 / (1.0 + (y * (y * 0.16666666666666666)))) / 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(Float64(x_m / Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666)))) / 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 / (1.0 + (y * (y * 0.16666666666666666)))) / 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[(N[(x$95$m / N[(1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.7%

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

   1. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{\frac{y}{\sin y}}\right), z\right) \]

   2. un-div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{\frac{y}{\sin y}}\right), z\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y}{\sin y}\right)\right), z\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \sin y\right)\right), z\right) \]

   5. sin-lowering-sin.f64 96.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{sin.f64}\left(y\right)\right)\right), z\right) \]

4. Applied egg-rr 96.7%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)}\right), z\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right), z\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right), z\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right)\right), z\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right)\right), z\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right)\right), z\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), z\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \frac{7}{360}\right)\right)\right)\right)\right)\right), z\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{7}{360}\right)\right)\right)\right)\right)\right), z\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{7}{360}\right)\right)\right)\right)\right)\right), z\right) \]

   10. *-lowering-*.f64 66.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{7}{360}\right)\right)\right)\right)\right)\right), z\right) \]

7. Simplified 66.1%

   \[\leadsto \frac{\frac{x}{\color{blue}{1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.019444444444444445\right)\right)}}}{z} \]

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right), z\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right)\right), z\right) \]

   2. *-lowering-*.f64 66.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right)\right), z\right) \]

10. Simplified 66.6%

    \[\leadsto \frac{\frac{x}{1 + y \cdot \color{blue}{\left(y \cdot 0.16666666666666666\right)}}}{z} \]
11. Add Preprocessing

Alternative 10: 66.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m}{z \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)} \end{array} \]

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 (+ 1.0 (* (* y y) 0.16666666666666666))))))

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 * (1.0 + ((y * y) * 0.16666666666666666))));
}

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 * (1.0d0 + ((y * y) * 0.16666666666666666d0))))
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 * (1.0 + ((y * y) * 0.16666666666666666))));
}

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 * (1.0 + ((y * y) * 0.16666666666666666))))

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 / Float64(z * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666)))))
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 * (1.0 + ((y * y) * 0.16666666666666666))));
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 / N[(z * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.7%

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

   1. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{\frac{y}{\sin y}}\right), z\right) \]

   2. un-div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{\frac{y}{\sin y}}\right), z\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y}{\sin y}\right)\right), z\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \sin y\right)\right), z\right) \]

   5. sin-lowering-sin.f64 96.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{sin.f64}\left(y\right)\right)\right), z\right) \]

4. Applied egg-rr 96.7%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)}\right), z\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right), z\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right), z\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right)\right), z\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right)\right), z\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right)\right), z\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), z\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \frac{7}{360}\right)\right)\right)\right)\right)\right), z\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{7}{360}\right)\right)\right)\right)\right)\right), z\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{7}{360}\right)\right)\right)\right)\right)\right), z\right) \]

   10. *-lowering-*.f64 66.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{7}{360}\right)\right)\right)\right)\right)\right), z\right) \]

7. Simplified 66.1%

   \[\leadsto \frac{\frac{x}{\color{blue}{1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.019444444444444445\right)\right)}}}{z} \]

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right), z\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right)\right), z\right) \]

   2. *-lowering-*.f64 66.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right)\right), z\right) \]

10. Simplified 66.6%

    \[\leadsto \frac{\frac{x}{1 + y \cdot \color{blue}{\left(y \cdot 0.16666666666666666\right)}}}{z} \]

11. Taylor expanded in x around 0

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

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(z \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)}\right) \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

    3. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{6}\right)\right)\right)\right) \]

    7. *-lowering-*.f64 66.6%

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{6}\right)\right)\right)\right) \]

13. Simplified 66.6%

    \[\leadsto \color{blue}{\frac{x}{z \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}} \]
14. Add Preprocessing

Alternative 11: 65.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \frac{x\_m}{z \cdot \left(-0.16666666666666666 \cdot \left(y \cdot y\right) + 1\right)} \end{array} \]

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 (+ (* -0.16666666666666666 (* y y)) 1.0)))))

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 * ((-0.16666666666666666 * (y * y)) + 1.0)));
}

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 * (((-0.16666666666666666d0) * (y * y)) + 1.0d0)))
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 * ((-0.16666666666666666 * (y * y)) + 1.0)));
}

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 * ((-0.16666666666666666 * (y * y)) + 1.0)))

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 / Float64(z * Float64(Float64(-0.16666666666666666 * Float64(y * y)) + 1.0))))
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 * ((-0.16666666666666666 * (y * y)) + 1.0)));
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 / N[(z * N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.7%

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

   1. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{\frac{y}{\sin y}}\right), z\right) \]

   2. un-div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{\frac{y}{\sin y}}\right), z\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y}{\sin y}\right)\right), z\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \sin y\right)\right), z\right) \]

   5. sin-lowering-sin.f64 96.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{sin.f64}\left(y\right)\right)\right), z\right) \]

4. Applied egg-rr 96.7%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)}\right), z\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right), z\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right), z\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right)\right), z\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right)\right), z\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{1}{6} + \frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right)\right), z\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{7}{360} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), z\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \frac{7}{360}\right)\right)\right)\right)\right)\right), z\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{7}{360}\right)\right)\right)\right)\right)\right), z\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{7}{360}\right)\right)\right)\right)\right)\right), z\right) \]

   10. *-lowering-*.f64 66.1%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{7}{360}\right)\right)\right)\right)\right)\right), z\right) \]

7. Simplified 66.1%

   \[\leadsto \frac{\frac{x}{\color{blue}{1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.019444444444444445\right)\right)}}}{z} \]

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} \cdot y\right)}\right)\right)\right), z\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{6}\right)\right)\right)\right), z\right) \]

   2. *-lowering-*.f64 66.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{6}\right)\right)\right)\right), z\right) \]

10. Simplified 66.6%

    \[\leadsto \frac{\frac{x}{1 + y \cdot \color{blue}{\left(y \cdot 0.16666666666666666\right)}}}{z} \]

12. Applied egg-rr 66.2%

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

13. Final simplification 66.2%

    \[\leadsto \frac{x}{z \cdot \left(-0.16666666666666666 \cdot \left(y \cdot y\right) + 1\right)} \]
14. Add Preprocessing

Alternative 12: 58.4% accurate, 35.7× 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 96.7%

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

3. Taylor expanded in y around 0

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

   1. /-lowering-/.f64 60.2%

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

5. Simplified 60.2%

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

Developer Target 1: 99.7% accurate, 0.9× 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 2024163 
(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))