Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.1% → 99.6%

Time: 11.5s

Alternatives: 15

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: 96.1% 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, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5e-80) (/ (/ x_m z) (/ y (sin y))) (/ (* x_m (/ (sin y) y)) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 5e-80) {
		tmp = (x_m / z) / (y / sin(y));
	} else {
		tmp = (x_m * (sin(y) / y)) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 5d-80) then
        tmp = (x_m / z) / (y / sin(y))
    else
        tmp = (x_m * (sin(y) / y)) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 5e-80) {
		tmp = (x_m / z) / (y / Math.sin(y));
	} else {
		tmp = (x_m * (Math.sin(y) / y)) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 5e-80:
		tmp = (x_m / z) / (y / math.sin(y))
	else:
		tmp = (x_m * (math.sin(y) / y)) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 5e-80)
		tmp = Float64(Float64(x_m / z) / Float64(y / sin(y)));
	else
		tmp = Float64(Float64(x_m * Float64(sin(y) / y)) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 5e-80)
		tmp = (x_m / z) / (y / sin(y));
	else
		tmp = (x_m * (sin(y) / y)) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 5e-80], N[(N[(x$95$m / z), $MachinePrecision] / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5e-80

   1. Initial program 94.7%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 95.2

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

   4. Applied egg-rr 95.2%

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

   if 5e-80 < x

   1. Initial program 99.8%

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

Alternative 2: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x\_m}{y}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+52}:\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{\sin y}{z \cdot y}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* x_m (/ (sin y) y)) z)))
   (*
    x_s
    (if (<= t_0 -5e-309)
      (* (/ (sin y) z) (/ x_m y))
      (if (<= t_0 4e+52)
        (/ (sin y) (* y (/ z x_m)))
        (* x_m (/ (sin y) (* z y))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -5e-309) {
		tmp = (sin(y) / z) * (x_m / y);
	} else if (t_0 <= 4e+52) {
		tmp = sin(y) / (y * (z / x_m));
	} else {
		tmp = x_m * (sin(y) / (z * y));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m * (sin(y) / y)) / z
    if (t_0 <= (-5d-309)) then
        tmp = (sin(y) / z) * (x_m / y)
    else if (t_0 <= 4d+52) then
        tmp = sin(y) / (y * (z / x_m))
    else
        tmp = x_m * (sin(y) / (z * y))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (Math.sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -5e-309) {
		tmp = (Math.sin(y) / z) * (x_m / y);
	} else if (t_0 <= 4e+52) {
		tmp = Math.sin(y) / (y * (z / x_m));
	} else {
		tmp = x_m * (Math.sin(y) / (z * y));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = (x_m * (math.sin(y) / y)) / z
	tmp = 0
	if t_0 <= -5e-309:
		tmp = (math.sin(y) / z) * (x_m / y)
	elif t_0 <= 4e+52:
		tmp = math.sin(y) / (y * (z / x_m))
	else:
		tmp = x_m * (math.sin(y) / (z * y))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m * Float64(sin(y) / y)) / z)
	tmp = 0.0
	if (t_0 <= -5e-309)
		tmp = Float64(Float64(sin(y) / z) * Float64(x_m / y));
	elseif (t_0 <= 4e+52)
		tmp = Float64(sin(y) / Float64(y * Float64(z / x_m)));
	else
		tmp = Float64(x_m * Float64(sin(y) / Float64(z * y)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (x_m * (sin(y) / y)) / z;
	tmp = 0.0;
	if (t_0 <= -5e-309)
		tmp = (sin(y) / z) * (x_m / y);
	elseif (t_0 <= 4e+52)
		tmp = sin(y) / (y * (z / x_m));
	else
		tmp = x_m * (sin(y) / (z * y));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -5e-309], N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+52], N[(N[Sin[y], $MachinePrecision] / N[(y * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -4.9999999999999995e-309

   1. Initial program 99.8%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. *-lft-identity N/A

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

      7. associate-*l/ N/A

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

      8. 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 83.5

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

   4. Applied egg-rr 83.5%

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

   if -4.9999999999999995e-309 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 4e52

   1. Initial program 92.4%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-/r/ N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/l/ N/A

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

      12. remove-double-div N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 N/A

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

      15. div-inv N/A

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

      16. remove-double-div N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 99.2

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

   4. Applied egg-rr 99.2%

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

   if 4e52 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.9%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.9

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

   4. Applied egg-rr 96.9%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 4 \cdot 10^{+52}:\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-137}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(x\_m \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{elif}\;t\_0 \leq 10^{-269}:\\ \;\;\;\;y \cdot \frac{x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* x_m (/ (sin y) y)) z)))
   (*
    x_s
    (if (<= t_0 -2e-137)
      (/ (* -0.16666666666666666 (* x_m (* y y))) z)
      (if (<= t_0 1e-269) (* y (/ x_m (* z y))) (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -2e-137) {
		tmp = (-0.16666666666666666 * (x_m * (y * y))) / z;
	} else if (t_0 <= 1e-269) {
		tmp = y * (x_m / (z * y));
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m * (sin(y) / y)) / z
    if (t_0 <= (-2d-137)) then
        tmp = ((-0.16666666666666666d0) * (x_m * (y * y))) / z
    else if (t_0 <= 1d-269) then
        tmp = y * (x_m / (z * y))
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (Math.sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -2e-137) {
		tmp = (-0.16666666666666666 * (x_m * (y * y))) / z;
	} else if (t_0 <= 1e-269) {
		tmp = y * (x_m / (z * y));
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = (x_m * (math.sin(y) / y)) / z
	tmp = 0
	if t_0 <= -2e-137:
		tmp = (-0.16666666666666666 * (x_m * (y * y))) / z
	elif t_0 <= 1e-269:
		tmp = y * (x_m / (z * y))
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m * Float64(sin(y) / y)) / z)
	tmp = 0.0
	if (t_0 <= -2e-137)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(x_m * Float64(y * y))) / z);
	elseif (t_0 <= 1e-269)
		tmp = Float64(y * Float64(x_m / Float64(z * y)));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (x_m * (sin(y) / y)) / z;
	tmp = 0.0;
	if (t_0 <= -2e-137)
		tmp = (-0.16666666666666666 * (x_m * (y * y))) / z;
	elseif (t_0 <= 1e-269)
		tmp = y * (x_m / (z * y));
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -2e-137], N[(N[(-0.16666666666666666 * N[(x$95$m * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 1e-269], N[(y * N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -1.99999999999999996e-137

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 60.7

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

   5. Simplified 60.7%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 4.8

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

   8. Simplified 4.8%

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

   if -1.99999999999999996e-137 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 9.9999999999999996e-270

   1. Initial program 90.7%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 84.0

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

   4. Applied egg-rr 84.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 47.9

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

   7. Simplified 47.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-rgt-identity N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-times N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 70.6

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 70.6

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

   9. Applied egg-rr 70.6%

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

   if 9.9999999999999996e-270 < (/.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 55.5

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

   5. Simplified 55.5%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -2 \cdot 10^{-137}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 10^{-269}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9997861905172747:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) 0.9997861905172747)
    (* (/ (sin y) z) (/ x_m y))
    (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 0.9997861905172747) {
		tmp = (sin(y) / z) * (x_m / y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((sin(y) / y) <= 0.9997861905172747d0) then
        tmp = (sin(y) / z) * (x_m / y)
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((Math.sin(y) / y) <= 0.9997861905172747) {
		tmp = (Math.sin(y) / z) * (x_m / y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (math.sin(y) / y) <= 0.9997861905172747:
		tmp = (math.sin(y) / z) * (x_m / y)
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.9997861905172747)
		tmp = Float64(Float64(sin(y) / z) * Float64(x_m / y));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((sin(y) / y) <= 0.9997861905172747)
		tmp = (sin(y) / z) * (x_m / y);
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.9997861905172747], N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99978619051727469

   1. Initial program 93.7%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. *-lft-identity N/A

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

      7. associate-*l/ N/A

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

      8. 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 93.8

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

   4. Applied egg-rr 93.8%

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

   if 0.99978619051727469 < (/.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. Simplified 100.0%

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

Alternative 5: 95.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9997861905172747:\\ \;\;\;\;\sin y \cdot \frac{x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) 0.9997861905172747)
    (* (sin y) (/ x_m (* z y)))
    (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 0.9997861905172747) {
		tmp = sin(y) * (x_m / (z * y));
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((sin(y) / y) <= 0.9997861905172747d0) then
        tmp = sin(y) * (x_m / (z * y))
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((Math.sin(y) / y) <= 0.9997861905172747) {
		tmp = Math.sin(y) * (x_m / (z * y));
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (math.sin(y) / y) <= 0.9997861905172747:
		tmp = math.sin(y) * (x_m / (z * y))
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.9997861905172747)
		tmp = Float64(sin(y) * Float64(x_m / Float64(z * y)));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((sin(y) / y) <= 0.9997861905172747)
		tmp = sin(y) * (x_m / (z * y));
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.9997861905172747], N[(N[Sin[y], $MachinePrecision] * N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99978619051727469

   1. Initial program 93.7%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. *-lft-identity N/A

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

      7. associate-*l/ N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-lft-identity N/A

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

      14. associate-/l/ N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 90.6

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

   4. Applied egg-rr 90.6%

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

   if 0.99978619051727469 < (/.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. Simplified 100.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9997861905172747:\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq 10^{-102}:\\ \;\;\;\;\frac{\frac{x\_m}{-z}}{\mathsf{fma}\left(y, y \cdot 0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (/ (sin y) y)) z) 1e-102)
    (/ (/ x_m (- z)) (fma y (* y 0.16666666666666666) -1.0))
    (*
     (/ x_m z)
     (fma
      y
      (* y (fma (* y y) 0.008333333333333333 -0.16666666666666666))
      1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (sin(y) / y)) / z) <= 1e-102) {
		tmp = (x_m / -z) / fma(y, (y * 0.16666666666666666), -1.0);
	} else {
		tmp = (x_m / z) * fma(y, (y * fma((y * y), 0.008333333333333333, -0.16666666666666666)), 1.0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= 1e-102)
		tmp = Float64(Float64(x_m / Float64(-z)) / fma(y, Float64(y * 0.16666666666666666), -1.0));
	else
		tmp = Float64(Float64(x_m / z) * fma(y, Float64(y * fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666)), 1.0));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e-102], N[(N[(x$95$m / (-z)), $MachinePrecision] / N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 9.99999999999999933e-103

   1. Initial program 95.4%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 95.9

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

   4. Applied egg-rr 95.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 63.5

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

   7. Simplified 63.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-/r* N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 63.5

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 63.5

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

   9. Applied egg-rr 63.5%

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

   11. Applied egg-rr 64.5%

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

   if 9.99999999999999933e-103 < (/.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. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 89.6

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

   4. Applied egg-rr 89.6%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. clear-num N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 87.3

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 87.3

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

   6. Applied egg-rr 87.3%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. +-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

   9. Simplified 56.9%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 10^{-102}:\\ \;\;\;\;\frac{\frac{x}{-z}}{\mathsf{fma}\left(y, y \cdot 0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq 10^{-269}:\\ \;\;\;\;\frac{\frac{x\_m}{-z}}{\mathsf{fma}\left(y, y \cdot 0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (/ (sin y) y)) z) 1e-269)
    (/ (/ x_m (- z)) (fma y (* y 0.16666666666666666) -1.0))
    (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (sin(y) / y)) / z) <= 1e-269) {
		tmp = (x_m / -z) / fma(y, (y * 0.16666666666666666), -1.0);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= 1e-269)
		tmp = Float64(Float64(x_m / Float64(-z)) / fma(y, Float64(y * 0.16666666666666666), -1.0));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e-269], N[(N[(x$95$m / (-z)), $MachinePrecision] / N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 9.9999999999999996e-270

   1. Initial program 94.9%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 95.4

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

   4. Applied egg-rr 95.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 65.1

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

   7. Simplified 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-/r* N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 65.2

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 65.2

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

   9. Applied egg-rr 65.2%

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

   11. Applied egg-rr 66.2%

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

   if 9.9999999999999996e-270 < (/.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 55.5

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

   5. Simplified 55.5%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{-z}}{\mathsf{fma}\left(y, y \cdot 0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq 10^{-269}:\\ \;\;\;\;\frac{\frac{x\_m}{\mathsf{fma}\left(y \cdot -0.16666666666666666, y, 1\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (/ (sin y) y)) z) 1e-269)
    (/ (/ x_m (fma (* y -0.16666666666666666) y 1.0)) z)
    (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (sin(y) / y)) / z) <= 1e-269) {
		tmp = (x_m / fma((y * -0.16666666666666666), y, 1.0)) / z;
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= 1e-269)
		tmp = Float64(Float64(x_m / fma(Float64(y * -0.16666666666666666), y, 1.0)) / z);
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e-269], N[(N[(x$95$m / N[(N[(y * -0.16666666666666666), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 9.9999999999999996e-270

   1. Initial program 94.9%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 95.4

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

   4. Applied egg-rr 95.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 65.1

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

   7. Simplified 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-/r* N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 65.2

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 65.2

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

   9. Applied egg-rr 65.2%

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

   11. Applied egg-rr 65.7%

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

   if 9.9999999999999996e-270 < (/.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 55.5

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

   5. Simplified 55.5%

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

Alternative 9: 65.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq 10^{-269}:\\ \;\;\;\;\frac{-x\_m}{z \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (/ (sin y) y)) z) 1e-269)
    (/ (- x_m) (* z (fma y (* y 0.16666666666666666) -1.0)))
    (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (sin(y) / y)) / z) <= 1e-269) {
		tmp = -x_m / (z * fma(y, (y * 0.16666666666666666), -1.0));
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= 1e-269)
		tmp = Float64(Float64(-x_m) / Float64(z * fma(y, Float64(y * 0.16666666666666666), -1.0)));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 1e-269], N[((-x$95$m) / N[(z * N[(y * N[(y * 0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 9.9999999999999996e-270

   1. Initial program 94.9%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 95.4

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

   4. Applied egg-rr 95.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 65.1

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

   7. Simplified 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-/r* N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 65.2

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 65.2

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

   9. Applied egg-rr 65.2%

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

   11. Applied egg-rr 65.7%

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

   if 9.9999999999999996e-270 < (/.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 55.5

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

   5. Simplified 55.5%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq 10^{-269}:\\ \;\;\;\;\frac{-x}{z \cdot \mathsf{fma}\left(y, y \cdot 0.16666666666666666, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq -2 \cdot 10^{-137}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(z, y \cdot \left(y \cdot 0.16666666666666666\right), z\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (/ (sin y) y)) z) -2e-137)
    (* (/ x_m z) (fma -0.16666666666666666 (* y y) 1.0))
    (/ x_m (fma z (* y (* y 0.16666666666666666)) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (sin(y) / y)) / z) <= -2e-137) {
		tmp = (x_m / z) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = x_m / fma(z, (y * (y * 0.16666666666666666)), z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= -2e-137)
		tmp = Float64(Float64(x_m / z) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(x_m / fma(z, Float64(y * Float64(y * 0.16666666666666666)), z));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], -2e-137], N[(N[(x$95$m / z), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(z * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -1.99999999999999996e-137

   1. Initial program 99.8%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 90.3

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

   4. Applied egg-rr 90.3%

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

   5. Taylor expanded in y around 0

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 60.7

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

   7. Simplified 60.7%

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

   if -1.99999999999999996e-137 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 95.0%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 96.2

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

   4. Applied egg-rr 96.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 63.2

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

   7. Simplified 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-/r* N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 62.7

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 62.7

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

   9. Applied egg-rr 62.7%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. associate-/l/ N/A

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

       4. lower-/.f64 N/A

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

       5. lift-fma.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*r* N/A

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

       9. lift-*.f64 N/A

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

       10. distribute-lft-in N/A

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

       11. *-rgt-identity N/A

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

       12. lower-fma.f64 N/A

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

       13. lower-*.f64 62.7

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

   11. Applied egg-rr 62.7%

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

Alternative 11: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \left(\sin y \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5.2e-80)
    (* (/ x_m z) (* (sin y) (/ 1.0 y)))
    (/ (* x_m (/ (sin y) y)) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 5.2e-80) {
		tmp = (x_m / z) * (sin(y) * (1.0 / y));
	} else {
		tmp = (x_m * (sin(y) / y)) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 5.2d-80) then
        tmp = (x_m / z) * (sin(y) * (1.0d0 / y))
    else
        tmp = (x_m * (sin(y) / y)) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 5.2e-80) {
		tmp = (x_m / z) * (Math.sin(y) * (1.0 / y));
	} else {
		tmp = (x_m * (Math.sin(y) / y)) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 5.2e-80:
		tmp = (x_m / z) * (math.sin(y) * (1.0 / y))
	else:
		tmp = (x_m * (math.sin(y) / y)) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 5.2e-80)
		tmp = Float64(Float64(x_m / z) * Float64(sin(y) * Float64(1.0 / y)));
	else
		tmp = Float64(Float64(x_m * Float64(sin(y) / y)) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 5.2e-80)
		tmp = (x_m / z) * (sin(y) * (1.0 / y));
	else
		tmp = (x_m * (sin(y) / y)) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 5.2e-80], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.2000000000000002e-80

   1. Initial program 94.7%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 95.2

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

   4. Applied egg-rr 95.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. clear-num N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 89.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 89.4

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

   6. Applied egg-rr 89.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. div-inv N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 95.1

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

   8. Applied egg-rr 95.1%

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

   if 5.2000000000000002e-80 < x

   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 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\sin y \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.8 \cdot 10^{-80}:\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 3.8e-80)
    (/ (sin y) (* y (/ z x_m)))
    (/ (* x_m (/ (sin y) y)) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 3.8e-80) {
		tmp = sin(y) / (y * (z / x_m));
	} else {
		tmp = (x_m * (sin(y) / y)) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 3.8d-80) then
        tmp = sin(y) / (y * (z / x_m))
    else
        tmp = (x_m * (sin(y) / y)) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 3.8e-80) {
		tmp = Math.sin(y) / (y * (z / x_m));
	} else {
		tmp = (x_m * (Math.sin(y) / y)) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 3.8e-80:
		tmp = math.sin(y) / (y * (z / x_m))
	else:
		tmp = (x_m * (math.sin(y) / y)) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 3.8e-80)
		tmp = Float64(sin(y) / Float64(y * Float64(z / x_m)));
	else
		tmp = Float64(Float64(x_m * Float64(sin(y) / y)) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 3.8e-80)
		tmp = sin(y) / (y * (z / x_m));
	else
		tmp = (x_m * (sin(y) / y)) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 3.8e-80], N[(N[Sin[y], $MachinePrecision] / N[(y * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.79999999999999967e-80

   1. Initial program 94.7%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-/r/ N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/l/ N/A

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

      12. remove-double-div N/A

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

      13. div-inv N/A

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

      14. lower-/.f64 N/A

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

      15. div-inv N/A

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

      16. remove-double-div N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 92.6

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

   4. Applied egg-rr 92.6%

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

   if 3.79999999999999967e-80 < x

   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 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-80}:\\ \;\;\;\;\frac{\sin y}{y \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{+32}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 2.9e+32)
    (* (/ x_m z) (fma -0.16666666666666666 (* y y) 1.0))
    (* y (/ x_m (* z y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 2.9e+32) {
		tmp = (x_m / z) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = y * (x_m / (z * y));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 2.9e+32)
		tmp = Float64(Float64(x_m / z) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(y * Float64(x_m / Float64(z * y)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 2.9e+32], N[(N[(x$95$m / z), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.90000000000000003e32

   1. Initial program 97.3%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 96.4

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

   4. Applied egg-rr 96.4%

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

   5. Taylor expanded in y around 0

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-lft-identity N/A

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

      7. distribute-rgt-out N/A

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

      8. +-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 65.8

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

   7. Simplified 65.8%

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

   if 2.90000000000000003e32 < y

   1. Initial program 94.1%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 90.3

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

   4. Applied egg-rr 90.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 26.5

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

   7. Simplified 26.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-rgt-identity N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-times N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 36.7

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 36.7

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

   9. Applied egg-rr 36.7%

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

Alternative 14: 61.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+42}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= y 2e+42) (/ x_m z) (* y (/ x_m (* z y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 2e+42) {
		tmp = x_m / z;
	} else {
		tmp = y * (x_m / (z * y));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2d+42) then
        tmp = x_m / z
    else
        tmp = y * (x_m / (z * y))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 2e+42) {
		tmp = x_m / z;
	} else {
		tmp = y * (x_m / (z * y));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= 2e+42:
		tmp = x_m / z
	else:
		tmp = y * (x_m / (z * y))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 2e+42)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(y * Float64(x_m / Float64(z * y)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= 2e+42)
		tmp = x_m / z;
	else
		tmp = y * (x_m / (z * y));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 2e+42], N[(x$95$m / z), $MachinePrecision], N[(y * N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000009e42

   1. Initial program 97.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 66.2

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

   5. Simplified 66.2%

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

   if 2.00000000000000009e42 < y

   1. Initial program 93.8%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 89.8

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

   4. Applied egg-rr 89.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 26.0

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

   7. Simplified 26.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-rgt-identity N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-times N/A

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

      14. *-rgt-identity N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 36.7

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 36.7

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

   9. Applied egg-rr 36.7%

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

Alternative 15: 57.9% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 53.6

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

5. Simplified 53.6%

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

Developer Target 1: 99.7% 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 2024219 
(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))