Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.0% → 99.6%

Time: 8.2s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{x}{\frac{z\_m}{\sin y} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y}{y} \cdot x}{z\_m}\\ \end{array} \end{array} \]

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 (<= z_m 1.5e-54)
    (/ x (* (/ z_m (sin y)) y))
    (/ (* (/ (sin y) 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 (z_m <= 1.5e-54) {
		tmp = x / ((z_m / sin(y)) * y);
	} else {
		tmp = ((sin(y) / y) * 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 (z_m <= 1.5d-54) then
        tmp = x / ((z_m / sin(y)) * y)
    else
        tmp = ((sin(y) / y) * 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 (z_m <= 1.5e-54) {
		tmp = x / ((z_m / Math.sin(y)) * y);
	} else {
		tmp = ((Math.sin(y) / y) * 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 z_m <= 1.5e-54:
		tmp = x / ((z_m / math.sin(y)) * y)
	else:
		tmp = ((math.sin(y) / y) * 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 (z_m <= 1.5e-54)
		tmp = Float64(x / Float64(Float64(z_m / sin(y)) * y));
	else
		tmp = Float64(Float64(Float64(sin(y) / y) * 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 (z_m <= 1.5e-54)
		tmp = x / ((z_m / sin(y)) * y);
	else
		tmp = ((sin(y) / y) * 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[z$95$m, 1.5e-54], N[(x / N[(N[(z$95$m / N[Sin[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.50000000000000005e-54

   1. Initial program 96.8%

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

      1. lift-/.f64 N/A

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

      2. remove-double-neg N/A

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

      3. lift-*.f64 N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. distribute-frac-neg2 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-/r/ N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 92.2

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

   4. Applied rewrites 92.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-*.f64 92.2

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

   6. Applied rewrites 92.2%

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

   if 1.50000000000000005e-54 < z

   1. Initial program 99.9%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{x}{\frac{z}{\sin y} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y}{y} \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999998888618363:\\ \;\;\;\;\frac{x}{y} \cdot \frac{\sin y}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \]

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) 0.9999998888618363)
    (* (/ x y) (/ (sin y) 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) <= 0.9999998888618363) {
		tmp = (x / y) * (sin(y) / 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) <= 0.9999998888618363d0) then
        tmp = (x / y) * (sin(y) / 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) <= 0.9999998888618363) {
		tmp = (x / y) * (Math.sin(y) / 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) <= 0.9999998888618363:
		tmp = (x / y) * (math.sin(y) / 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(sin(y) / y) <= 0.9999998888618363)
		tmp = Float64(Float64(x / y) * Float64(sin(y) / 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) <= 0.9999998888618363)
		tmp = (x / y) * (sin(y) / 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[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.9999998888618363], N[(N[(x / y), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.999999888861836328

   1. Initial program 95.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 95.9

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

   4. Applied rewrites 95.9%

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

   if 0.999999888861836328 < (/.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. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999998888618363:\\ \;\;\;\;\frac{x}{y} \cdot \frac{\sin y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.3% accurate, 0.8× speedup

Math

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

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) 0.0)
    (* (* y x) (/ 1.0 (* y 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) <= 0.0) {
		tmp = (y * x) * (1.0 / (y * 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) <= 0.0d0) then
        tmp = (y * x) * (1.0d0 / (y * 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) <= 0.0) {
		tmp = (y * x) * (1.0 / (y * 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) <= 0.0:
		tmp = (y * x) * (1.0 / (y * 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) <= 0.0)
		tmp = Float64(Float64(y * x) * Float64(1.0 / Float64(y * 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) <= 0.0)
		tmp = (y * x) * (1.0 / (y * 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], 0.0], N[(N[(y * x), $MachinePrecision] * N[(1.0 / N[(y * z$95$m), $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) < 0.0

   1. Initial program 96.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. div-inv N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 85.0

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 49.8

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

   7. Applied rewrites 49.8%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 56.5

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

   5. Applied rewrites 56.5%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq 0:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{1}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 5 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{z\_m} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \]

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) 5e-140) (* (/ 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) <= 5e-140) {
		tmp = (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) <= 5d-140) then
        tmp = (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) <= 5e-140) {
		tmp = (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) <= 5e-140:
		tmp = (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) <= 5e-140)
		tmp = Float64(Float64(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) <= 5e-140)
		tmp = (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], 5e-140], N[(N[(y / 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) < 5.00000000000000015e-140

   1. Initial program 95.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 95.5

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 22.9

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

   7. Applied rewrites 22.9%

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

   if 5.00000000000000015e-140 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 82.8

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

   5. Applied rewrites 82.8%

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

Alternative 5: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y \cdot z\_m} \cdot x\\ \end{array} \end{array} \]

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 (<= y 2e-8) (/ x z_m) (* (/ (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 (y <= 2e-8) {
		tmp = x / z_m;
	} 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 (y <= 2d-8) then
        tmp = x / z_m
    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 (y <= 2e-8) {
		tmp = x / z_m;
	} 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 y <= 2e-8:
		tmp = x / z_m
	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 (y <= 2e-8)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(Float64(sin(y) / Float64(y * 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 (y <= 2e-8)
		tmp = x / z_m;
	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[y, 2e-8], N[(x / z$95$m), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2e-8

   1. Initial program 97.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 73.8

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

   5. Applied rewrites 73.8%

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

   if 2e-8 < y

   1. Initial program 97.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 91.4

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

   4. Applied rewrites 91.4%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y \cdot z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z\_m} \cdot \sin y\\ \end{array} \end{array} \]

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 (<= y 1.7e-40) (/ x z_m) (* (/ x (* y z_m)) (sin y)))))

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 (y <= 1.7e-40) {
		tmp = x / z_m;
	} else {
		tmp = (x / (y * z_m)) * sin(y);
	}
	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 (y <= 1.7d-40) then
        tmp = x / z_m
    else
        tmp = (x / (y * z_m)) * sin(y)
    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 (y <= 1.7e-40) {
		tmp = x / z_m;
	} else {
		tmp = (x / (y * z_m)) * Math.sin(y);
	}
	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 y <= 1.7e-40:
		tmp = x / z_m
	else:
		tmp = (x / (y * z_m)) * math.sin(y)
	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 (y <= 1.7e-40)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(Float64(x / Float64(y * z_m)) * sin(y));
	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 (y <= 1.7e-40)
		tmp = x / z_m;
	else
		tmp = (x / (y * z_m)) * sin(y);
	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[y, 1.7e-40], N[(x / z$95$m), $MachinePrecision], N[(N[(x / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.69999999999999992e-40

   1. Initial program 97.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 72.6

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

   5. Applied rewrites 72.6%

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

   if 1.69999999999999992e-40 < y

   1. Initial program 98.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. associate-/l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 92.1

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

   4. Applied rewrites 92.1%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-40}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \sin y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 6.4:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z\_m} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

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 (<= y 6.4)
    (* (fma -0.16666666666666666 (* y y) 1.0) (/ x z_m))
    (* (/ y z_m) (/ x y)))))

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 (y <= 6.4) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * (x / z_m);
	} else {
		tmp = (y / z_m) * (x / y);
	}
	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 (y <= 6.4)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * Float64(x / z_m));
	else
		tmp = Float64(Float64(y / z_m) * Float64(x / y));
	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[y, 6.4], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y / z$95$m), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.4000000000000004

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.1

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

   5. Applied rewrites 70.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 70.5

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

   7. Applied rewrites 70.5%

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

   if 6.4000000000000004 < y

   1. Initial program 97.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 21.9

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

   7. Applied rewrites 21.9%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.4:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.4% accurate, 4.0× speedup

Math

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

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) (- (fma (* (* 0.16666666666666666 z_m) y) y 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 / -fma(((0.16666666666666666 * z_m) * y), y, 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(Float64(-x) / Float64(-fma(Float64(Float64(0.16666666666666666 * z_m) * y), y, 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) / (-N[(N[(N[(0.16666666666666666 * z$95$m), $MachinePrecision] * y), $MachinePrecision] * y + z$95$m), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.9%

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

   1. lift-/.f64 N/A

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

   2. remove-double-neg N/A

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

   3. lift-*.f64 N/A

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

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

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

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

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

   6. associate-*r/ N/A

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

   7. clear-num N/A

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

   8. div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. distribute-frac-neg2 N/A

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

   12. lift-/.f64 N/A

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

   13. associate-/r/ N/A

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

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

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-neg.f64 88.7

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

4. Applied rewrites 88.7%

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

5. Taylor expanded in y around 0

   \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{-1 \cdot z + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right)}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right)} \]

   2. neg-sub0 N/A

      \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\left(0 - z\right)} + {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right)} \]

   3. associate-+l- N/A

      \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{0 - \left(z - {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right)\right)}} \]

   4. sub0-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\mathsf{neg}\left(\left(z - {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right)\right)\right)}} \]

   5. lower-neg.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\mathsf{neg}\left(\left(z - {y}^{2} \cdot \left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right)\right)\right)}} \]

   6. *-commutative N/A

      \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(\left(z - \color{blue}{\left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right) \cdot {y}^{2}}\right)\right)} \]

   7. cancel-sign-sub-inv N/A

      \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(\color{blue}{\left(z + \left(\mathsf{neg}\left(\left(\frac{-1}{6} \cdot z + {y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right)\right)\right) \cdot {y}^{2}\right)}\right)} \]

   8. +-commutative N/A

      \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(\left(z + \left(\mathsf{neg}\left(\color{blue}{\left({y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right) + \frac{-1}{6} \cdot z\right)}\right)\right) \cdot {y}^{2}\right)\right)} \]

   9. distribute-neg-out N/A

      \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(\left(z + \color{blue}{\left(\left(\mathsf{neg}\left({y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right)\right) + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot z\right)\right)\right)} \cdot {y}^{2}\right)\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(\left(z + \left(\color{blue}{-1 \cdot \left({y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right)} + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot z\right)\right)\right) \cdot {y}^{2}\right)\right)} \]

   11. sub-neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(\left(z + \color{blue}{\left(-1 \cdot \left({y}^{2} \cdot \left(\frac{-1}{36} \cdot z + \frac{1}{120} \cdot z\right)\right) - \frac{-1}{6} \cdot z\right)} \cdot {y}^{2}\right)\right)} \]

7. Applied rewrites 63.8%

   \[\leadsto \frac{-x}{\color{blue}{-\mathsf{fma}\left(\left(z \cdot \mathsf{fma}\left(0.019444444444444445, y \cdot y, 0.16666666666666666\right)\right) \cdot y, y, z\right)}} \]

8. Taylor expanded in y around 0

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

10. Applied rewrites 64.0%

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

11. Final simplification 64.0%

    \[\leadsto \frac{-x}{-\mathsf{fma}\left(\left(0.16666666666666666 \cdot z\right) \cdot y, y, z\right)} \]
12. Add Preprocessing

Alternative 9: 66.4% accurate, 4.6× speedup

Math

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

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 (fma (* (* y y) z_m) 0.16666666666666666 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 / fma(((y * y) * z_m), 0.16666666666666666, 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 / fma(Float64(Float64(y * y) * z_m), 0.16666666666666666, 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 / N[(N[(N[(y * y), $MachinePrecision] * z$95$m), $MachinePrecision] * 0.16666666666666666 + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.9%

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

   1. lift-/.f64 N/A

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

   2. remove-double-neg N/A

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

   3. lift-*.f64 N/A

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

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

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

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

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

   6. associate-*r/ N/A

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

   7. clear-num N/A

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

   8. div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. distribute-frac-neg2 N/A

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

   12. lift-/.f64 N/A

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

   13. associate-/r/ N/A

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

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

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lower-neg.f64 88.7

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

4. Applied rewrites 88.7%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f64 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-neg.f64 N/A

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

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

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

   9. remove-double-neg N/A

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

   10. lower-*.f64 88.7

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

6. Applied rewrites 88.7%

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

7. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 63.9

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

9. Applied rewrites 63.9%

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

Alternative 10: 59.1% 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 97.9%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 57.6

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

5. Applied rewrites 57.6%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
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 2024240 
(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))