Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 95.9% → 98.9%

Time: 11.7s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{-116}:\\ \;\;\;\;\frac{t\_0}{z\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot x}{z\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (* z_s (if (<= z_m 2e-116) (* (/ t_0 z_m) x) (/ (* t_0 x) z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double t_0 = sin(y) / y;
	double tmp;
	if (z_m <= 2e-116) {
		tmp = (t_0 / z_m) * x;
	} else {
		tmp = (t_0 * x) / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (z_m <= 2d-116) then
        tmp = (t_0 / z_m) * x
    else
        tmp = (t_0 * x) / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (z_m <= 2e-116) {
		tmp = (t_0 / z_m) * x;
	} else {
		tmp = (t_0 * x) / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	t_0 = math.sin(y) / y
	tmp = 0
	if z_m <= 2e-116:
		tmp = (t_0 / z_m) * x
	else:
		tmp = (t_0 * x) / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (z_m <= 2e-116)
		tmp = Float64(Float64(t_0 / z_m) * x);
	else
		tmp = Float64(Float64(t_0 * x) / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (z_m <= 2e-116)
		tmp = (t_0 / z_m) * x;
	else
		tmp = (t_0 * x) / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(z$95$s * If[LessEqual[z$95$m, 2e-116], N[(N[(t$95$0 / z$95$m), $MachinePrecision] * x), $MachinePrecision], N[(N[(t$95$0 * x), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2e-116

   1. Initial program 93.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 88.4

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

   4. Applied egg-rr 88.4%

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

      1. associate-/r* N/A

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

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

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

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

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

      4. sin-lowering-sin.f64 96.5

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

   6. Applied egg-rr 96.5%

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

   if 2e-116 < z

   1. Initial program 99.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{\sin y}{y}}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y}{y} \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \cdot x \leq -2 \cdot 10^{-188}:\\ \;\;\;\;\frac{\sin y}{z\_m} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{z\_m} \cdot x\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (*
    z_s
    (if (<= (* t_0 x) -2e-188)
      (* (/ (sin y) z_m) (/ x y))
      (* (/ t_0 z_m) x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double t_0 = sin(y) / y;
	double tmp;
	if ((t_0 * x) <= -2e-188) {
		tmp = (sin(y) / z_m) * (x / y);
	} else {
		tmp = (t_0 / z_m) * x;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if ((t_0 * x) <= (-2d-188)) then
        tmp = (sin(y) / z_m) * (x / y)
    else
        tmp = (t_0 / z_m) * x
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if ((t_0 * x) <= -2e-188) {
		tmp = (Math.sin(y) / z_m) * (x / y);
	} else {
		tmp = (t_0 / z_m) * x;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	t_0 = math.sin(y) / y
	tmp = 0
	if (t_0 * x) <= -2e-188:
		tmp = (math.sin(y) / z_m) * (x / y)
	else:
		tmp = (t_0 / z_m) * x
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (Float64(t_0 * x) <= -2e-188)
		tmp = Float64(Float64(sin(y) / z_m) * Float64(x / y));
	else
		tmp = Float64(Float64(t_0 / z_m) * x);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if ((t_0 * x) <= -2e-188)
		tmp = (sin(y) / z_m) * (x / y);
	else
		tmp = (t_0 / z_m) * x;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(z$95$s * If[LessEqual[N[(t$95$0 * x), $MachinePrecision], -2e-188], N[(N[(N[Sin[y], $MachinePrecision] / z$95$m), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / z$95$m), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -1.9999999999999999e-188

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-*l/ N/A

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

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

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

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

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

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

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. /-lowering-/.f64 87.5

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

   4. Applied egg-rr 87.5%

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

   if -1.9999999999999999e-188 < (*.f64 x (/.f64 (sin.f64 y) y))

   1. Initial program 92.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 91.6

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

   4. Applied egg-rr 91.6%

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

      1. associate-/r* N/A

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

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

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

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

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

      4. sin-lowering-sin.f64 96.7

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

   6. Applied egg-rr 96.7%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \cdot x \leq -2 \cdot 10^{-188}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y}{y}}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999999995:\\ \;\;\;\;\frac{\sin y}{z\_m} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (sin y) y) 0.9999999995) (* (/ (sin y) z_m) (/ x y)) (/ x z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 0.9999999995) {
		tmp = (sin(y) / z_m) * (x / y);
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((sin(y) / y) <= 0.9999999995d0) then
        tmp = (sin(y) / z_m) * (x / y)
    else
        tmp = x / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((Math.sin(y) / y) <= 0.9999999995) {
		tmp = (Math.sin(y) / z_m) * (x / y);
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if (math.sin(y) / y) <= 0.9999999995:
		tmp = (math.sin(y) / z_m) * (x / y)
	else:
		tmp = x / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.9999999995)
		tmp = Float64(Float64(sin(y) / z_m) * Float64(x / y));
	else
		tmp = Float64(x / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if ((sin(y) / y) <= 0.9999999995)
		tmp = (sin(y) / z_m) * (x / y);
	else
		tmp = x / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.9999999995], N[(N[(N[Sin[y], $MachinePrecision] / z$95$m), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99999999949999996

   1. Initial program 90.0%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-*l/ N/A

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

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

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

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

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

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

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. /-lowering-/.f64 90.1

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

   4. Applied egg-rr 90.1%

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

   if 0.99999999949999996 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

Alternative 4: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999999995:\\ \;\;\;\;\frac{\sin y \cdot x}{z\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (sin y) y) 0.9999999995) (/ (* (sin y) x) (* z_m y)) (/ x z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 0.9999999995) {
		tmp = (sin(y) * x) / (z_m * y);
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((sin(y) / y) <= 0.9999999995d0) then
        tmp = (sin(y) * x) / (z_m * y)
    else
        tmp = x / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((Math.sin(y) / y) <= 0.9999999995) {
		tmp = (Math.sin(y) * x) / (z_m * y);
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if (math.sin(y) / y) <= 0.9999999995:
		tmp = (math.sin(y) * x) / (z_m * y)
	else:
		tmp = x / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.9999999995)
		tmp = Float64(Float64(sin(y) * x) / Float64(z_m * y));
	else
		tmp = Float64(x / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if ((sin(y) / y) <= 0.9999999995)
		tmp = (sin(y) * x) / (z_m * y);
	else
		tmp = x / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.9999999995], N[(N[(N[Sin[y], $MachinePrecision] * x), $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99999999949999996

   1. Initial program 90.0%

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 88.5

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

   4. Applied egg-rr 88.5%

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

   if 0.99999999949999996 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999999995:\\ \;\;\;\;\frac{\sin y \cdot x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999999995:\\ \;\;\;\;\sin y \cdot \frac{x}{z\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (sin y) y) 0.9999999995) (* (sin y) (/ x (* z_m y))) (/ x z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 0.9999999995) {
		tmp = sin(y) * (x / (z_m * y));
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((sin(y) / y) <= 0.9999999995d0) then
        tmp = sin(y) * (x / (z_m * y))
    else
        tmp = x / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((Math.sin(y) / y) <= 0.9999999995) {
		tmp = Math.sin(y) * (x / (z_m * y));
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if (math.sin(y) / y) <= 0.9999999995:
		tmp = math.sin(y) * (x / (z_m * y))
	else:
		tmp = x / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.9999999995)
		tmp = Float64(sin(y) * Float64(x / Float64(z_m * y)));
	else
		tmp = Float64(x / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if ((sin(y) / y) <= 0.9999999995)
		tmp = sin(y) * (x / (z_m * y));
	else
		tmp = x / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.9999999995], N[(N[Sin[y], $MachinePrecision] * N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99999999949999996

   1. Initial program 90.0%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*l/ N/A

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

      8. *-lft-identity N/A

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

      9. associate-/l/ N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. sin-lowering-sin.f64 88.4

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

   4. Applied egg-rr 88.4%

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

   if 0.99999999949999996 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999999995:\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z\_m} \leq 10^{-312}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \frac{1}{x - y \cdot \left(x \cdot \left(y \cdot -0.16666666666666666\right)\right)}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (* (/ (sin y) y) x) z_m) 1e-312)
    (/ (* (* x x) (/ 1.0 (- x (* y (* x (* y -0.16666666666666666)))))) z_m)
    (/ x z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((((sin(y) / y) * x) / z_m) <= 1e-312) {
		tmp = ((x * x) * (1.0 / (x - (y * (x * (y * -0.16666666666666666)))))) / z_m;
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((((sin(y) / y) * x) / z_m) <= 1d-312) then
        tmp = ((x * x) * (1.0d0 / (x - (y * (x * (y * (-0.16666666666666666d0))))))) / z_m
    else
        tmp = x / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((((Math.sin(y) / y) * x) / z_m) <= 1e-312) {
		tmp = ((x * x) * (1.0 / (x - (y * (x * (y * -0.16666666666666666)))))) / z_m;
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if (((math.sin(y) / y) * x) / z_m) <= 1e-312:
		tmp = ((x * x) * (1.0 / (x - (y * (x * (y * -0.16666666666666666)))))) / z_m
	else:
		tmp = x / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(sin(y) / y) * x) / z_m) <= 1e-312)
		tmp = Float64(Float64(Float64(x * x) * Float64(1.0 / Float64(x - Float64(y * Float64(x * Float64(y * -0.16666666666666666)))))) / z_m);
	else
		tmp = Float64(x / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if ((((sin(y) / y) * x) / z_m) <= 1e-312)
		tmp = ((x * x) * (1.0 / (x - (y * (x * (y * -0.16666666666666666)))))) / z_m;
	else
		tmp = x / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] / z$95$m), $MachinePrecision], 1e-312], N[(N[(N[(x * x), $MachinePrecision] * N[(1.0 / N[(x - N[(y * N[(x * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 9.9999999999847e-313

   1. Initial program 91.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \frac{x \cdot {y}^{2}}{z} + \frac{x}{z}} \]
   4. 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. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r* N/A

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

      14. /-lowering-/.f64 N/A

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

   5. Simplified 49.6%

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

      1. flip-+ N/A

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

      2. div-inv N/A

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

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

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

   7. Applied egg-rr 32.1%

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

   8. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

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

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

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

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

      5. neg-lowering-neg.f64 54.0

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

   10. Simplified 54.0%

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

   if 9.9999999999847e-313 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 59.4

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

   5. Simplified 59.4%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq 10^{-312}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \frac{1}{x - y \cdot \left(x \cdot \left(y \cdot -0.16666666666666666\right)\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-5}:\\ \;\;\;\;\frac{y}{\frac{z\_m \cdot y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (sin y) y) 1e-5)
    (/ y (/ (* z_m y) x))
    (/
     (*
      x
      (fma
       (* y y)
       (fma (* y y) 0.008333333333333333 -0.16666666666666666)
       1.0))
     z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 1e-5) {
		tmp = y / ((z_m * y) / x);
	} else {
		tmp = (x * fma((y * y), fma((y * y), 0.008333333333333333, -0.16666666666666666), 1.0)) / z_m;
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 1e-5)
		tmp = Float64(y / Float64(Float64(z_m * y) / x));
	else
		tmp = Float64(Float64(x * fma(Float64(y * y), fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), 1.0)) / z_m);
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 1e-5], N[(y / N[(N[(z$95$m * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 1.00000000000000008e-5

   1. Initial program 89.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 88.2

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

   4. Applied egg-rr 88.2%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 16.2%

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

      1. div-inv N/A

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

      2. associate-*l* N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. un-div-inv N/A

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

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

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 22.5

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

   9. Applied egg-rr 22.5%

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

   if 1.00000000000000008e-5 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

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

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

      5. sub-neg N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 99.3

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

   5. Simplified 99.3%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-5}:\\ \;\;\;\;\frac{y}{\frac{z \cdot y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z\_m} \leq 2 \cdot 10^{-273}:\\ \;\;\;\;y \cdot \frac{x}{z\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (* (/ (sin y) y) x) z_m) 2e-273)
    (* y (/ x (* z_m y)))
    (/ x z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((((sin(y) / y) * x) / z_m) <= 2e-273) {
		tmp = y * (x / (z_m * y));
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((((sin(y) / y) * x) / z_m) <= 2d-273) then
        tmp = y * (x / (z_m * y))
    else
        tmp = x / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((((Math.sin(y) / y) * x) / z_m) <= 2e-273) {
		tmp = y * (x / (z_m * y));
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if (((math.sin(y) / y) * x) / z_m) <= 2e-273:
		tmp = y * (x / (z_m * y))
	else:
		tmp = x / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(sin(y) / y) * x) / z_m) <= 2e-273)
		tmp = Float64(y * Float64(x / Float64(z_m * y)));
	else
		tmp = Float64(x / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if ((((sin(y) / y) * x) / z_m) <= 2e-273)
		tmp = y * (x / (z_m * y));
	else
		tmp = x / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] / z$95$m), $MachinePrecision], 2e-273], N[(y * N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 2e-273

   1. Initial program 92.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 88.5

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

   4. Applied egg-rr 88.5%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 52.0%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. times-frac N/A

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

      5. /-rgt-identity N/A

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

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

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

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

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

      8. *-lowering-*.f64 53.1

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

   9. Applied egg-rr 53.1%

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

   if 2e-273 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 60.7

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

   5. Simplified 60.7%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq 2 \cdot 10^{-273}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.02:\\ \;\;\;\;\frac{y}{\frac{z\_m \cdot y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (sin y) y) 0.02)
    (/ y (/ (* z_m y) x))
    (* (/ x z_m) (fma y (* y -0.16666666666666666) 1.0)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 0.02) {
		tmp = y / ((z_m * y) / x);
	} else {
		tmp = (x / z_m) * fma(y, (y * -0.16666666666666666), 1.0);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.02)
		tmp = Float64(y / Float64(Float64(z_m * y) / x));
	else
		tmp = Float64(Float64(x / z_m) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.02], N[(y / N[(N[(z$95$m * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.0200000000000000004

   1. Initial program 90.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 88.3

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

   4. Applied egg-rr 88.3%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 16.2%

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

      1. div-inv N/A

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

      2. associate-*l* N/A

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

      3. associate-/r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. un-div-inv N/A

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

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

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 22.5

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

   9. Applied egg-rr 22.5%

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

   if 0.0200000000000000004 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 86.7

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

   4. Applied egg-rr 86.7%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

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

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

      11. /-lowering-/.f64 99.7

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

   7. Simplified 99.7%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

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

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

      11. /-lowering-/.f64 100.0

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.02:\\ \;\;\;\;\frac{y}{\frac{z \cdot y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.02:\\ \;\;\;\;\frac{y}{y \cdot \frac{z\_m}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (sin y) y) 0.02)
    (/ y (* y (/ z_m x)))
    (* (/ x z_m) (fma y (* y -0.16666666666666666) 1.0)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 0.02) {
		tmp = y / (y * (z_m / x));
	} else {
		tmp = (x / z_m) * fma(y, (y * -0.16666666666666666), 1.0);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.02)
		tmp = Float64(y / Float64(y * Float64(z_m / x)));
	else
		tmp = Float64(Float64(x / z_m) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.02], N[(y / N[(y * N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.0200000000000000004

   1. Initial program 90.0%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. associate-/l/ N/A

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

      5. remove-double-div N/A

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

      6. div-inv N/A

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

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

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

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

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

      9. div-inv N/A

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

      10. remove-double-div N/A

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

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

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

      12. /-lowering-/.f64 94.4

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

   4. Applied egg-rr 94.4%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 22.2%

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

   if 0.0200000000000000004 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 86.7

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

   4. Applied egg-rr 86.7%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

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

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

      11. /-lowering-/.f64 99.7

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

   7. Simplified 99.7%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

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

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

      11. /-lowering-/.f64 100.0

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.02:\\ \;\;\;\;\frac{y}{y \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.02:\\ \;\;\;\;y \cdot \frac{x}{z\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (sin y) y) 0.02)
    (* y (/ x (* z_m y)))
    (* (/ x z_m) (fma y (* y -0.16666666666666666) 1.0)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 0.02) {
		tmp = y * (x / (z_m * y));
	} else {
		tmp = (x / z_m) * fma(y, (y * -0.16666666666666666), 1.0);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.02)
		tmp = Float64(y * Float64(x / Float64(z_m * y)));
	else
		tmp = Float64(Float64(x / z_m) * fma(y, Float64(y * -0.16666666666666666), 1.0));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.02], N[(y * N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.0200000000000000004

   1. Initial program 90.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 88.3

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

   4. Applied egg-rr 88.3%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 16.2%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. times-frac N/A

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

      5. /-rgt-identity N/A

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

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

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

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

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

      8. *-lowering-*.f64 22.2

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

   9. Applied egg-rr 22.2%

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

   if 0.0200000000000000004 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 86.7

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

   4. Applied egg-rr 86.7%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

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

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

      11. /-lowering-/.f64 99.7

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

   7. Simplified 99.7%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

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

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

      11. /-lowering-/.f64 100.0

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.02:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{\sin y}{y}}{z\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z\_m}{x}}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 5e+84) (* (/ (/ (sin y) y) z_m) x) (/ (sin y) (* y (/ z_m x))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 5e+84) {
		tmp = ((sin(y) / y) / z_m) * x;
	} else {
		tmp = sin(y) / (y * (z_m / x));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 5d+84) then
        tmp = ((sin(y) / y) / z_m) * x
    else
        tmp = sin(y) / (y * (z_m / x))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 5e+84) {
		tmp = ((Math.sin(y) / y) / z_m) * x;
	} else {
		tmp = Math.sin(y) / (y * (z_m / x));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 5e+84:
		tmp = ((math.sin(y) / y) / z_m) * x
	else:
		tmp = math.sin(y) / (y * (z_m / x))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 5e+84)
		tmp = Float64(Float64(Float64(sin(y) / y) / z_m) * x);
	else
		tmp = Float64(sin(y) / Float64(y * Float64(z_m / x)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 5e+84)
		tmp = ((sin(y) / y) / z_m) * x;
	else
		tmp = sin(y) / (y * (z_m / x));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 5e+84], N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] / z$95$m), $MachinePrecision] * x), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] / N[(y * N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.0000000000000001e84

   1. Initial program 94.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 89.2

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

   4. Applied egg-rr 89.2%

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

      1. associate-/r* N/A

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

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

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

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

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

      4. sin-lowering-sin.f64 96.7

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

   6. Applied egg-rr 96.7%

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

   if 5.0000000000000001e84 < z

   1. Initial program 99.8%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. associate-/l/ N/A

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

      5. remove-double-div N/A

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

      6. div-inv N/A

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

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

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

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

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

      9. div-inv N/A

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

      10. remove-double-div N/A

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

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

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

      12. /-lowering-/.f64 96.5

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

   4. Applied egg-rr 96.5%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{\sin y}{y}}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.0% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * N[(x / z$95$m), $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. /-lowering-/.f64 56.7

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

5. Simplified 56.7%

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

Developer Target 1: 99.5% 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 2024198 
(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))