Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.0% → 99.6%

Time: 12.4s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-36}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m \cdot \sin y}{y}}{z}\\ \end{array} \end{array} \]

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 1e-36) (* (/ (sin y) y) (/ x_m z)) (/ (/ (* 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 <= 1e-36) {
		tmp = (sin(y) / y) * (x_m / z);
	} 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 <= 1d-36) then
        tmp = (sin(y) / y) * (x_m / z)
    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 <= 1e-36) {
		tmp = (Math.sin(y) / y) * (x_m / z);
	} 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 <= 1e-36:
		tmp = (math.sin(y) / y) * (x_m / z)
	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 <= 1e-36)
		tmp = Float64(Float64(sin(y) / y) * Float64(x_m / z));
	else
		tmp = Float64(Float64(Float64(x_m * 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 <= 1e-36)
		tmp = (sin(y) / y) * (x_m / z);
	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, 1e-36], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * N[Sin[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999994e-37

   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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.9

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

   4. Applied egg-rr 95.9%

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

   if 9.9999999999999994e-37 < x

   1. Initial program 99.6%

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

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

      7. clear-num N/A

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

      8. clear-num N/A

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

      9. associate-*r/ N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 99.6

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r/ N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 99.8

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

   4. Applied egg-rr 99.8%

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

Alternative 2: 62.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-254}:\\ \;\;\;\;\frac{-1}{\frac{z \cdot z}{z} \cdot \frac{-1}{x\_m}}\\ \mathbf{elif}\;t\_0 \leq 10^{-316}:\\ \;\;\;\;\frac{1}{\frac{x\_m + \frac{x\_m}{z} \cdot 0}{x\_m \cdot \frac{x\_m}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* x_m (/ (sin y) y))))
   (*
    x_s
    (if (<= t_0 -4e-254)
      (/ -1.0 (* (/ (* z z) z) (/ -1.0 x_m)))
      (if (<= t_0 1e-316)
        (/ 1.0 (/ (+ x_m (* (/ x_m z) 0.0)) (* x_m (/ x_m z))))
        (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = x_m * (sin(y) / y);
	double tmp;
	if (t_0 <= -4e-254) {
		tmp = -1.0 / (((z * z) / z) * (-1.0 / x_m));
	} else if (t_0 <= 1e-316) {
		tmp = 1.0 / ((x_m + ((x_m / z) * 0.0)) / (x_m * (x_m / z)));
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_m * (sin(y) / y)
    if (t_0 <= (-4d-254)) then
        tmp = (-1.0d0) / (((z * z) / z) * ((-1.0d0) / x_m))
    else if (t_0 <= 1d-316) then
        tmp = 1.0d0 / ((x_m + ((x_m / z) * 0.0d0)) / (x_m * (x_m / z)))
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = x_m * (Math.sin(y) / y);
	double tmp;
	if (t_0 <= -4e-254) {
		tmp = -1.0 / (((z * z) / z) * (-1.0 / x_m));
	} else if (t_0 <= 1e-316) {
		tmp = 1.0 / ((x_m + ((x_m / z) * 0.0)) / (x_m * (x_m / z)));
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = x_m * (math.sin(y) / y)
	tmp = 0
	if t_0 <= -4e-254:
		tmp = -1.0 / (((z * z) / z) * (-1.0 / x_m))
	elif t_0 <= 1e-316:
		tmp = 1.0 / ((x_m + ((x_m / z) * 0.0)) / (x_m * (x_m / z)))
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(x_m * Float64(sin(y) / y))
	tmp = 0.0
	if (t_0 <= -4e-254)
		tmp = Float64(-1.0 / Float64(Float64(Float64(z * z) / z) * Float64(-1.0 / x_m)));
	elseif (t_0 <= 1e-316)
		tmp = Float64(1.0 / Float64(Float64(x_m + Float64(Float64(x_m / z) * 0.0)) / Float64(x_m * Float64(x_m / z))));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = x_m * (sin(y) / y);
	tmp = 0.0;
	if (t_0 <= -4e-254)
		tmp = -1.0 / (((z * z) / z) * (-1.0 / x_m));
	elseif (t_0 <= 1e-316)
		tmp = 1.0 / ((x_m + ((x_m / z) * 0.0)) / (x_m * (x_m / z)));
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -4e-254], N[(-1.0 / N[(N[(N[(z * z), $MachinePrecision] / z), $MachinePrecision] * N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-316], N[(1.0 / N[(N[(x$95$m + N[(N[(x$95$m / z), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -3.9999999999999996e-254

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 60.2

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

   5. Simplified 60.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 60.0

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

   7. Applied egg-rr 60.0%

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

      1. lift-/.f64 N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 59.7

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

   9. Applied egg-rr 59.7%

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

       1. neg-sub0 N/A

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

       2. flip-- N/A

          \[\leadsto \frac{-1}{\frac{1}{x} \cdot \color{blue}{\frac{0 \cdot 0 - z \cdot z}{0 + z}}} \]

       3. +-lft-identity N/A

          \[\leadsto \frac{-1}{\frac{1}{x} \cdot \frac{0 \cdot 0 - z \cdot z}{\color{blue}{z}}} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{-1}{\frac{1}{x} \cdot \color{blue}{\frac{0 \cdot 0 - z \cdot z}{z}}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{-1}{\frac{1}{x} \cdot \frac{\color{blue}{0} - z \cdot z}{z}} \]

       6. sub0-neg N/A

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

       7. lower-neg.f64 N/A

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

       8. lower-*.f64 44.6

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

   11. Applied egg-rr 44.6%

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

   if -3.9999999999999996e-254 < (*.f64 x (/.f64 (sin.f64 y) y)) < 9.999999837e-317

   1. Initial program 73.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 32.6

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

   5. Simplified 32.6%

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

      1. clear-num N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 32.5

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

   7. Applied egg-rr 32.5%

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

      1. frac-2neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{0 - z}}{\mathsf{neg}\left(x\right)}} \]

      3. div-sub N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{0}{\mathsf{neg}\left(x\right)} - \frac{z}{\mathsf{neg}\left(x\right)}}} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto \frac{1}{\frac{0}{\mathsf{neg}\left(x\right)} - \color{blue}{\left(\mathsf{neg}\left(\frac{z}{x}\right)\right)}} \]

      5. clear-num N/A

         \[\leadsto \frac{1}{\frac{0}{\mathsf{neg}\left(x\right)} - \left(\mathsf{neg}\left(\color{blue}{\frac{1}{\frac{x}{z}}}\right)\right)} \]

      6. distribute-neg-frac N/A

         \[\leadsto \frac{1}{\frac{0}{\mathsf{neg}\left(x\right)} - \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\frac{x}{z}}}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{0}{\mathsf{neg}\left(x\right)} - \frac{\color{blue}{-1}}{\frac{x}{z}}} \]

      8. frac-sub N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{0 \cdot \frac{x}{z} - \left(\mathsf{neg}\left(x\right)\right) \cdot -1}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{x}{z}}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{0 \cdot \frac{x}{z} - \left(\mathsf{neg}\left(x\right)\right) \cdot -1}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{x}{z}}}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{0 \cdot \frac{x}{z} - \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{\frac{1}{-1}}}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{x}{z}}} \]

      11. div-inv N/A

          \[\leadsto \frac{1}{\frac{0 \cdot \frac{x}{z} - \color{blue}{\frac{\mathsf{neg}\left(x\right)}{-1}}}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{x}{z}}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{0 \cdot \frac{x}{z} - \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\mathsf{neg}\left(1\right)}}}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{x}{z}}} \]

      13. frac-2neg N/A

          \[\leadsto \frac{1}{\frac{0 \cdot \frac{x}{z} - \color{blue}{\frac{x}{1}}}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{x}{z}}} \]

      14. /-rgt-identity N/A

          \[\leadsto \frac{1}{\frac{0 \cdot \frac{x}{z} - \color{blue}{x}}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{x}{z}}} \]

      15. lower--.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{0 \cdot \frac{x}{z} - x}}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{x}{z}}} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{0 \cdot \frac{x}{z}} - x}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{x}{z}}} \]

      17. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{0 \cdot \color{blue}{\frac{x}{z}} - x}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{x}{z}}} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{0 \cdot \frac{x}{z} - x}{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{x}{z}}}} \]

      19. lower-neg.f64 N/A

          \[\leadsto \frac{1}{\frac{0 \cdot \frac{x}{z} - x}{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \frac{x}{z}}} \]

      20. lower-/.f64 44.4

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

   9. Applied egg-rr 44.4%

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

   if 9.999999837e-317 < (*.f64 x (/.f64 (sin.f64 y) y))

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 66.0

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

   5. Simplified 66.0%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq -4 \cdot 10^{-254}:\\ \;\;\;\;\frac{-1}{\frac{z \cdot z}{z} \cdot \frac{-1}{x}}\\ \mathbf{elif}\;x \cdot \frac{\sin y}{y} \leq 10^{-316}:\\ \;\;\;\;\frac{1}{\frac{x + \frac{x}{z} \cdot 0}{x \cdot \frac{x}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := x\_m \cdot \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-318}:\\ \;\;\;\;\frac{-1}{\frac{z \cdot z}{z} \cdot \frac{-1}{x\_m}}\\ \mathbf{elif}\;t\_0 \leq 10^{-250}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \end{array} \]

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))))
   (*
    x_s
    (if (<= t_0 -4e-318)
      (/ -1.0 (* (/ (* z z) z) (/ -1.0 x_m)))
      (if (<= t_0 1e-250)
        (* (/ y z) (/ x_m y))
        (*
         (/ 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 t_0 = x_m * (sin(y) / y);
	double tmp;
	if (t_0 <= -4e-318) {
		tmp = -1.0 / (((z * z) / z) * (-1.0 / x_m));
	} else if (t_0 <= 1e-250) {
		tmp = (y / z) * (x_m / y);
	} 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)
	t_0 = Float64(x_m * Float64(sin(y) / y))
	tmp = 0.0
	if (t_0 <= -4e-318)
		tmp = Float64(-1.0 / Float64(Float64(Float64(z * z) / z) * Float64(-1.0 / x_m)));
	elseif (t_0 <= 1e-250)
		tmp = Float64(Float64(y / z) * Float64(x_m / y));
	else
		tmp = Float64(Float64(x_m / z) * fma(y, Float64(y * fma(y, Float64(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_] := Block[{t$95$0 = N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -4e-318], N[(-1.0 / N[(N[(N[(z * z), $MachinePrecision] / z), $MachinePrecision] * N[(-1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-250], N[(N[(y / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (/.f64 (sin.f64 y) y)) < -3.9999999e-318

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 58.9

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

   5. Simplified 58.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 58.8

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

   7. Applied egg-rr 58.8%

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

      1. lift-/.f64 N/A

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

      2. /-rgt-identity N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. metadata-eval N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 58.5

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

   9. Applied egg-rr 58.5%

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

       1. neg-sub0 N/A

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

       2. flip-- N/A

          \[\leadsto \frac{-1}{\frac{1}{x} \cdot \color{blue}{\frac{0 \cdot 0 - z \cdot z}{0 + z}}} \]

       3. +-lft-identity N/A

          \[\leadsto \frac{-1}{\frac{1}{x} \cdot \frac{0 \cdot 0 - z \cdot z}{\color{blue}{z}}} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{-1}{\frac{1}{x} \cdot \color{blue}{\frac{0 \cdot 0 - z \cdot z}{z}}} \]

       5. metadata-eval N/A

          \[\leadsto \frac{-1}{\frac{1}{x} \cdot \frac{\color{blue}{0} - z \cdot z}{z}} \]

       6. sub0-neg N/A

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

       7. lower-neg.f64 N/A

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

       8. lower-*.f64 45.0

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

   11. Applied egg-rr 45.0%

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

   if -3.9999999e-318 < (*.f64 x (/.f64 (sin.f64 y) y)) < 1.0000000000000001e-250

   1. Initial program 73.1%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. *-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 72.9

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

   4. Applied egg-rr 72.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 58.2

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

   7. Simplified 58.2%

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

   if 1.0000000000000001e-250 < (*.f64 x (/.f64 (sin.f64 y) y))

   1. Initial program 99.7%

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

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 91.2

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

   4. Applied egg-rr 91.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 64.9

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

   7. Simplified 64.9%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{\sin y}{y} \leq -4 \cdot 10^{-318}:\\ \;\;\;\;\frac{-1}{\frac{z \cdot z}{z} \cdot \frac{-1}{x}}\\ \mathbf{elif}\;x \cdot \frac{\sin y}{y} \leq 10^{-250}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 0.5× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (*
    x_s
    (if (<= (/ (* x_m t_0) z) -5e-15)
      (/ x_m (/ (* y z) (sin y)))
      (* t_0 (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (((x_m * t_0) / z) <= -5e-15) {
		tmp = x_m / ((y * z) / sin(y));
	} else {
		tmp = t_0 * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (((x_m * t_0) / z) <= (-5d-15)) then
        tmp = x_m / ((y * z) / sin(y))
    else
        tmp = t_0 * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (((x_m * t_0) / z) <= -5e-15) {
		tmp = x_m / ((y * z) / Math.sin(y));
	} else {
		tmp = t_0 * (x_m / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if ((x_m * t_0) / z) <= -5e-15:
		tmp = x_m / ((y * z) / math.sin(y))
	else:
		tmp = t_0 * (x_m / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (Float64(Float64(x_m * t_0) / z) <= -5e-15)
		tmp = Float64(x_m / Float64(Float64(y * z) / sin(y)));
	else
		tmp = Float64(t_0 * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (((x_m * t_0) / z) <= -5e-15)
		tmp = x_m / ((y * z) / sin(y));
	else
		tmp = t_0 * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(x$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision], -5e-15], N[(x$95$m / N[(N[(y * z), $MachinePrecision] / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -4.99999999999999999e-15

   1. Initial program 99.7%

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

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

      8. associate-/r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 78.1

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

   4. Applied egg-rr 78.1%

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

   if -4.99999999999999999e-15 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 94.8%

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

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.0

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

   4. Applied egg-rr 96.0%

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

Alternative 5: 94.4% accurate, 0.5× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (*
    x_s
    (if (<= (/ (* x_m t_0) z) -5e-15)
      (* (* x_m (sin y)) (/ 1.0 (* y z)))
      (* t_0 (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (((x_m * t_0) / z) <= -5e-15) {
		tmp = (x_m * sin(y)) * (1.0 / (y * z));
	} else {
		tmp = t_0 * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (((x_m * t_0) / z) <= (-5d-15)) then
        tmp = (x_m * sin(y)) * (1.0d0 / (y * z))
    else
        tmp = t_0 * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (((x_m * t_0) / z) <= -5e-15) {
		tmp = (x_m * Math.sin(y)) * (1.0 / (y * z));
	} else {
		tmp = t_0 * (x_m / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if ((x_m * t_0) / z) <= -5e-15:
		tmp = (x_m * math.sin(y)) * (1.0 / (y * z))
	else:
		tmp = t_0 * (x_m / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (Float64(Float64(x_m * t_0) / z) <= -5e-15)
		tmp = Float64(Float64(x_m * sin(y)) * Float64(1.0 / Float64(y * z)));
	else
		tmp = Float64(t_0 * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (((x_m * t_0) / z) <= -5e-15)
		tmp = (x_m * sin(y)) * (1.0 / (y * z));
	else
		tmp = t_0 * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(x$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision], -5e-15], N[(N[(x$95$m * N[Sin[y], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -4.99999999999999999e-15

   1. Initial program 99.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. clear-num N/A

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

      4. clear-num N/A

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

      5. div-inv N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 77.6

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

   4. Applied egg-rr 77.6%

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

   if -4.99999999999999999e-15 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 94.8%

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

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.0

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

   4. Applied egg-rr 96.0%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -5 \cdot 10^{-15}:\\ \;\;\;\;\left(x \cdot \sin y\right) \cdot \frac{1}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.99999999:\\ \;\;\;\;\sin y \cdot \frac{x\_m}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \end{array} \]

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 (<= (/ (sin y) y) 0.99999999)
    (* (sin y) (/ x_m (* y z)))
    (* (/ x_m z) (fma -0.16666666666666666 (* y y) 1.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 0.99999999) {
		tmp = sin(y) * (x_m / (y * z));
	} else {
		tmp = (x_m / z) * fma(-0.16666666666666666, (y * y), 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(sin(y) / y) <= 0.99999999)
		tmp = Float64(sin(y) * Float64(x_m / Float64(y * z)));
	else
		tmp = Float64(Float64(x_m / z) * fma(-0.16666666666666666, Float64(y * y), 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[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.99999999], N[(N[Sin[y], $MachinePrecision] * N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99999998999999995

   1. Initial program 91.9%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. *-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 92.4

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

   4. Applied egg-rr 92.4%

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

   if 0.99999998999999995 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 100.0

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

   7. Simplified 100.0%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.99999999:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.05:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \end{array} \end{array} \]

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 (<= (/ (sin y) y) 0.05)
    (* (/ y z) (/ x_m y))
    (*
     (/ 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 ((sin(y) / y) <= 0.05) {
		tmp = (y / z) * (x_m / y);
	} 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(sin(y) / y) <= 0.05)
		tmp = Float64(Float64(y / z) * Float64(x_m / y));
	else
		tmp = Float64(Float64(x_m / z) * fma(y, Float64(y * fma(y, Float64(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[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.05], N[(N[(y / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.050000000000000003

   1. Initial program 91.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 91.7

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

   4. Applied egg-rr 91.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 22.0

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

   7. Simplified 22.0%

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

   if 0.050000000000000003 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 100.0

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

   7. Simplified 100.0%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.05:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.8% accurate, 0.9× speedup

Math

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

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 (<= (* x_m (/ (sin y) y)) 0.0) (* (/ 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 ((x_m * (sin(y) / y)) <= 0.0) {
		tmp = (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 ((x_m * (sin(y) / y)) <= 0.0d0) then
        tmp = (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 ((x_m * (Math.sin(y) / y)) <= 0.0) {
		tmp = (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 (x_m * (math.sin(y) / y)) <= 0.0:
		tmp = (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(x_m * Float64(sin(y) / y)) <= 0.0)
		tmp = Float64(Float64(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 ((x_m * (sin(y) / y)) <= 0.0)
		tmp = (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[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(y / 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 x (/.f64 (sin.f64 y) y)) < -0.0

   1. Initial program 93.2%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 83.7

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

   4. Applied egg-rr 83.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 48.5

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

   7. Simplified 48.5%

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

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

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 64.9

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

   5. Simplified 64.9%

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

Alternative 9: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 9 \cdot 10^{-32}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot t\_0}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (* x_s (if (<= x_m 9e-32) (* t_0 (/ x_m z)) (/ (* x_m t_0) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (x_m <= 9e-32) {
		tmp = t_0 * (x_m / z);
	} else {
		tmp = (x_m * t_0) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (x_m <= 9d-32) then
        tmp = t_0 * (x_m / z)
    else
        tmp = (x_m * t_0) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (x_m <= 9e-32) {
		tmp = t_0 * (x_m / z);
	} else {
		tmp = (x_m * t_0) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if x_m <= 9e-32:
		tmp = t_0 * (x_m / z)
	else:
		tmp = (x_m * t_0) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (x_m <= 9e-32)
		tmp = Float64(t_0 * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m * t_0) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (x_m <= 9e-32)
		tmp = t_0 * (x_m / z);
	else
		tmp = (x_m * t_0) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 9e-32], N[(t$95$0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.00000000000000009e-32

   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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.0

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

   4. Applied egg-rr 96.0%

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

   if 9.00000000000000009e-32 < x

   1. Initial program 99.6%

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

Alternative 10: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \sin y}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 9.2e-7)
    (* (/ x_m z) (fma -0.16666666666666666 (* y y) 1.0))
    (/ (* 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 (y <= 9.2e-7) {
		tmp = (x_m / z) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = (x_m * 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 (y <= 9.2e-7)
		tmp = Float64(Float64(x_m / z) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(x_m * sin(y)) / Float64(y * 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[y, 9.2e-7], N[(N[(x$95$m / z), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[Sin[y], $MachinePrecision]), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.1999999999999998e-7

   1. Initial program 96.5%

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

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.1

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

   4. Applied egg-rr 95.1%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 67.3

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

   7. Simplified 67.3%

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

   if 9.1999999999999998e-7 < y

   1. Initial program 93.9%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. 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 91.7

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

   4. Applied egg-rr 91.7%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sin y}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 860000000:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(y, y \cdot -0.16666666666666666, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x\_m}{y}\\ \end{array} \end{array} \]

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 860000000.0)
    (/ (* x_m (fma y (* y -0.16666666666666666) 1.0)) z)
    (* (/ y z) (/ x_m 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 <= 860000000.0) {
		tmp = (x_m * fma(y, (y * -0.16666666666666666), 1.0)) / z;
	} else {
		tmp = (y / z) * (x_m / 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 <= 860000000.0)
		tmp = Float64(Float64(x_m * fma(y, Float64(y * -0.16666666666666666), 1.0)) / z);
	else
		tmp = Float64(Float64(y / z) * Float64(x_m / 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, 860000000.0], N[(N[(x$95$m * N[(y * N[(y * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 8.6e8

   1. Initial program 96.5%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 66.0

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

   5. Simplified 66.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. flip-+ N/A

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

      4. lift-*.f64 N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

   7. Applied egg-rr 37.5%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 66.0

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

   10. Simplified 66.0%

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

   if 8.6e8 < y

   1. Initial program 93.6%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 93.7

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

   4. Applied egg-rr 93.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 19.2

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

   7. Simplified 19.2%

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

Alternative 12: 57.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 46000000000:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x\_m}{y}\\ \end{array} \end{array} \]

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 46000000000.0)
    (* (/ x_m z) (fma -0.16666666666666666 (* y y) 1.0))
    (* (/ y z) (/ x_m 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 <= 46000000000.0) {
		tmp = (x_m / z) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = (y / z) * (x_m / 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 <= 46000000000.0)
		tmp = Float64(Float64(x_m / z) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(y / z) * Float64(x_m / 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, 46000000000.0], N[(N[(x$95$m / z), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.6e10

   1. Initial program 96.5%

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

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.1

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

   4. Applied egg-rr 95.1%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 66.9

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

   7. Simplified 66.9%

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

   if 4.6e10 < y

   1. Initial program 93.6%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 93.7

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

   4. Applied egg-rr 93.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 19.2

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

   7. Simplified 19.2%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 46000000000:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 860000000:\\ \;\;\;\;x\_m \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x\_m}{y}\\ \end{array} \end{array} \]

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 860000000.0)
    (* x_m (/ (fma -0.16666666666666666 (* y y) 1.0) z))
    (* (/ y z) (/ x_m 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 <= 860000000.0) {
		tmp = x_m * (fma(-0.16666666666666666, (y * y), 1.0) / z);
	} else {
		tmp = (y / z) * (x_m / 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 <= 860000000.0)
		tmp = Float64(x_m * Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) / z));
	else
		tmp = Float64(Float64(y / z) * Float64(x_m / 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, 860000000.0], N[(x$95$m * N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 8.6e8

   1. Initial program 96.5%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 83.2

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

   4. Applied egg-rr 83.2%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

      14. associate-/l* N/A

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

   7. Simplified 65.8%

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

   if 8.6e8 < y

   1. Initial program 93.6%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 93.7

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

   4. Applied egg-rr 93.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 19.2

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

   7. Simplified 19.2%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 860000000:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.4% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 58.6

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

5. Simplified 58.6%

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

Developer Target 1: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\sin y}\\ t_1 := \frac{x \cdot \frac{1}{t\_0}}{z}\\ \mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;\frac{x}{z \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (sin y))) (t_1 (/ (* x (/ 1.0 t_0)) z)))
   (if (< z -4.2173720203427147e-29)
     t_1
     (if (< z 4.446702369113811e+64) (/ x (* z t_0)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = y / sin(y);
	double t_1 = (x * (1.0 / t_0)) / z;
	double tmp;
	if (z < -4.2173720203427147e-29) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x / (z * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y / sin(y)
    t_1 = (x * (1.0d0 / t_0)) / z
    if (z < (-4.2173720203427147d-29)) then
        tmp = t_1
    else if (z < 4.446702369113811d+64) then
        tmp = x / (z * t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / Math.sin(y);
	double t_1 = (x * (1.0 / t_0)) / z;
	double tmp;
	if (z < -4.2173720203427147e-29) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x / (z * t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / math.sin(y)
	t_1 = (x * (1.0 / t_0)) / z
	tmp = 0
	if z < -4.2173720203427147e-29:
		tmp = t_1
	elif z < 4.446702369113811e+64:
		tmp = x / (z * t_0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / sin(y))
	t_1 = Float64(Float64(x * Float64(1.0 / t_0)) / z)
	tmp = 0.0
	if (z < -4.2173720203427147e-29)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = Float64(x / Float64(z * t_0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / sin(y);
	t_1 = (x * (1.0 / t_0)) / z;
	tmp = 0.0;
	if (z < -4.2173720203427147e-29)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = x / (z * t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Less[z, -4.2173720203427147e-29], t$95$1, If[Less[z, 4.446702369113811e+64], N[(x / N[(z * t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024212 
(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))