Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.1% → 99.8%

Time: 8.2s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y \cdot x\_m}{y}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5.5e-8)
    (* (/ x_m z) (/ (sin y) y))
    (/ (/ (* (sin y) x_m) y) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 5.5e-8) {
		tmp = (x_m / z) * (sin(y) / y);
	} else {
		tmp = ((sin(y) * x_m) / y) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 5.5d-8) then
        tmp = (x_m / z) * (sin(y) / y)
    else
        tmp = ((sin(y) * x_m) / y) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 5.5e-8) {
		tmp = (x_m / z) * (Math.sin(y) / y);
	} else {
		tmp = ((Math.sin(y) * x_m) / y) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 5.5e-8:
		tmp = (x_m / z) * (math.sin(y) / y)
	else:
		tmp = ((math.sin(y) * x_m) / y) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 5.5e-8)
		tmp = Float64(Float64(x_m / z) * Float64(sin(y) / y));
	else
		tmp = Float64(Float64(Float64(sin(y) * x_m) / y) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 5.5e-8)
		tmp = (x_m / z) * (sin(y) / y);
	else
		tmp = ((sin(y) * x_m) / y) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 5.5e-8], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[y], $MachinePrecision] * x$95$m), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.5000000000000003e-8

   1. Initial program 97.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* 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 95.5

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

   4. Applied rewrites 95.5%

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

   if 5.5000000000000003e-8 < x

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y \cdot x}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9998:\\ \;\;\;\;\frac{\sin y}{z \cdot y} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right) \cdot x\_m, y \cdot y, x\_m\right)}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) 0.9998)
    (* (/ (sin y) (* z y)) x_m)
    (/
     (fma
      (*
       (fma
        (fma -0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        -0.16666666666666666)
       x_m)
      (* y y)
      x_m)
     z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 0.9998) {
		tmp = (sin(y) / (z * y)) * x_m;
	} else {
		tmp = fma((fma(fma(-0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), -0.16666666666666666) * x_m), (y * y), x_m) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.9998)
		tmp = Float64(Float64(sin(y) / Float64(z * y)) * x_m);
	else
		tmp = Float64(fma(Float64(fma(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), -0.16666666666666666) * x_m), Float64(y * y), x_m) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.9998], N[(N[(N[Sin[y], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(y * y), $MachinePrecision] + x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99980000000000002

   1. Initial program 95.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 95.3

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

   4. Applied rewrites 95.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 91.1

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

   6. Applied rewrites 91.1%

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

   if 0.99980000000000002 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x + {y}^{2} \cdot \left(\frac{-1}{6} \cdot x + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{120} \cdot x\right)\right)}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot x + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{120} \cdot x\right)\right) + x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot x + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{120} \cdot x\right)\right) \cdot {y}^{2}} + x}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot x + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{120} \cdot x\right), {y}^{2}, x\right)}}{z} \]

   5. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, x\right)}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9998:\\ \;\;\;\;\frac{\sin y}{z \cdot y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right) \cdot x, y \cdot y, x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999:\\ \;\;\;\;\frac{\sin y \cdot x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) 0.9999)
    (/ (* (sin y) x_m) (* z y))
    (* (fma -0.16666666666666666 (* y y) 1.0) (/ x_m z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 0.9999) {
		tmp = (sin(y) * x_m) / (z * y);
	} else {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * (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)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.9999)
		tmp = Float64(Float64(sin(y) * x_m) / Float64(z * y));
	else
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.9999], N[(N[(N[Sin[y], $MachinePrecision] * x$95$m), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99990000000000001

   1. Initial program 95.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 91.1

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

   4. Applied rewrites 91.1%

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

   if 0.99990000000000001 < (/.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-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* 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 rewrites 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. Applied rewrites 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. Add Preprocessing

Alternative 4: 95.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{x\_m}{z \cdot y} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right) \cdot x\_m, y \cdot y, x\_m\right)}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) 5e-12)
    (* (/ x_m (* z y)) (sin y))
    (/
     (fma
      (*
       (fma
        (fma -0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        -0.16666666666666666)
       x_m)
      (* y y)
      x_m)
     z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 5e-12) {
		tmp = (x_m / (z * y)) * sin(y);
	} else {
		tmp = fma((fma(fma(-0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), -0.16666666666666666) * x_m), (y * y), x_m) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 5e-12)
		tmp = Float64(Float64(x_m / Float64(z * y)) * sin(y));
	else
		tmp = Float64(fma(Float64(fma(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), -0.16666666666666666) * x_m), Float64(y * y), x_m) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 5e-12], N[(N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(y * y), $MachinePrecision] + x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 4.9999999999999997e-12

   1. Initial program 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 95.3

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

   4. Applied rewrites 95.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 91.0

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

   6. Applied rewrites 91.0%

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

   8. Applied rewrites 91.0%

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

   if 4.9999999999999997e-12 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{x + {y}^{2} \cdot \left(\frac{-1}{6} \cdot x + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{120} \cdot x\right)\right)}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{{y}^{2} \cdot \left(\frac{-1}{6} \cdot x + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{120} \cdot x\right)\right) + x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{6} \cdot x + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{120} \cdot x\right)\right) \cdot {y}^{2}} + x}{z} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot x + {y}^{2} \cdot \left(\frac{-1}{5040} \cdot \left(x \cdot {y}^{2}\right) + \frac{1}{120} \cdot x\right), {y}^{2}, x\right)}}{z} \]

   5. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, x\right)}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{z \cdot y} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right) \cdot x, y \cdot y, x\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 2 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{y \cdot x\_m}{-z}}{y \cdot y} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y \cdot y, 0.008333333333333333, -0.16666666666666666\right), y \cdot y, 1\right)}{\frac{z}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) 2e-84)
    (* (/ (/ (* y x_m) (- z)) (* y y)) y)
    (/
     (fma (fma (* y y) 0.008333333333333333 -0.16666666666666666) (* y y) 1.0)
     (/ z x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 2e-84) {
		tmp = (((y * x_m) / -z) / (y * y)) * y;
	} else {
		tmp = fma(fma((y * y), 0.008333333333333333, -0.16666666666666666), (y * y), 1.0) / (z / x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 2e-84)
		tmp = Float64(Float64(Float64(Float64(y * x_m) / Float64(-z)) / Float64(y * y)) * y);
	else
		tmp = Float64(fma(fma(Float64(y * y), 0.008333333333333333, -0.16666666666666666), Float64(y * y), 1.0) / Float64(z / x_m));
	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], 2e-84], N[(N[(N[(N[(y * x$95$m), $MachinePrecision] / (-z)), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 2.0000000000000001e-84

   1. Initial program 94.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 91.5

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

   4. Applied rewrites 91.5%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 21.4

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

   7. Applied rewrites 21.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. frac-2neg-rev N/A

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

      5. neg-sub0 N/A

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

      6. flip-- N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. neg-sub0 N/A

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

      10. +-lft-identity N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. neg-mul-1 N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. clear-num N/A

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

      18. frac-2neg N/A

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

      19. metadata-eval N/A

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

      20. lift-neg.f64 N/A

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

      21. inv-pow N/A

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

      22. pow-flip N/A

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

      25. sqrt-pow1 N/A

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

      26. pow2 N/A

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

      27. lift-neg.f64 N/A

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

      28. lift-neg.f64 N/A

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

      29. sqr-neg-rev N/A

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

   9. Applied rewrites 37.2%

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

   if 2.0000000000000001e-84 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 91.7

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

   5. Applied rewrites 91.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{Rewrite=>}\left(lower-fma.f64, \left(\mathsf{fma}\left(y \cdot y, \frac{1}{120}, \frac{-1}{6}\right)\right)\right), y \cdot y, 1\right)}{\color{blue}{\frac{z}{x}}} \]

   7. Applied rewrites 92.3%

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

Alternative 6: 65.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 5 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{y \cdot x\_m}{-z}}{y \cdot y} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) 5e-105)
    (* (/ (/ (* y x_m) (- z)) (* y y)) y)
    (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= 5e-105) {
		tmp = (((y * x_m) / -z) / (y * y)) * y;
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((sin(y) / y) <= 5d-105) then
        tmp = (((y * x_m) / -z) / (y * y)) * y
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((Math.sin(y) / y) <= 5e-105) {
		tmp = (((y * x_m) / -z) / (y * y)) * y;
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (math.sin(y) / y) <= 5e-105:
		tmp = (((y * x_m) / -z) / (y * y)) * y
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 5e-105)
		tmp = Float64(Float64(Float64(Float64(y * x_m) / Float64(-z)) / Float64(y * y)) * y);
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((sin(y) / y) <= 5e-105)
		tmp = (((y * x_m) / -z) / (y * y)) * y;
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 5e-105], N[(N[(N[(N[(y * x$95$m), $MachinePrecision] / (-z)), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 4.99999999999999963e-105

   1. Initial program 94.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 92.1

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 21.1

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

   7. Applied rewrites 21.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. frac-2neg-rev N/A

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

      5. neg-sub0 N/A

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

      6. flip-- N/A

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

      7. metadata-eval N/A

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

      8. lift-*.f64 N/A

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

      9. neg-sub0 N/A

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

      10. +-lft-identity N/A

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

      11. distribute-neg-frac N/A

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

      12. distribute-neg-frac2 N/A

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

      13. neg-mul-1 N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. div-inv N/A

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

      17. clear-num N/A

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

      18. frac-2neg N/A

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

      19. metadata-eval N/A

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

      20. lift-neg.f64 N/A

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

      21. inv-pow N/A

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

      22. pow-flip N/A

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

      25. sqrt-pow1 N/A

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

      26. pow2 N/A

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

      27. lift-neg.f64 N/A

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

      28. lift-neg.f64 N/A

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

      29. sqr-neg-rev N/A

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

   9. Applied rewrites 37.5%

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

   if 4.99999999999999963e-105 < (/.f64 (sin.f64 y) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 90.6

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

   5. Applied rewrites 90.6%

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

Alternative 7: 39.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x\_m}{z} \leq 10^{-319}:\\ \;\;\;\;\frac{y \cdot x\_m}{\left(-z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* (/ (sin y) y) x_m) z) 1e-319)
    (/ (* y x_m) (* (- z) y))
    (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((((sin(y) / y) * x_m) / z) <= 1e-319) {
		tmp = (y * x_m) / (-z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((((sin(y) / y) * x_m) / z) <= 1d-319) then
        tmp = (y * x_m) / (-z * y)
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((((Math.sin(y) / y) * x_m) / z) <= 1e-319) {
		tmp = (y * x_m) / (-z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (((math.sin(y) / y) * x_m) / z) <= 1e-319:
		tmp = (y * x_m) / (-z * y)
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(sin(y) / y) * x_m) / z) <= 1e-319)
		tmp = Float64(Float64(y * x_m) / Float64(Float64(-z) * y));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((((sin(y) / y) * x_m) / z) <= 1e-319)
		tmp = (y * x_m) / (-z * y);
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], 1e-319], N[(N[(y * x$95$m), $MachinePrecision] / N[((-z) * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 9.99989e-320

   1. Initial program 97.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 81.7

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

   4. Applied rewrites 81.7%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 47.0

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

   7. Applied rewrites 47.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-neg-out N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 47.0

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

      9. lift-*.f64 N/A

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

      10. unpow1 N/A

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

      11. metadata-eval N/A

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

      12. sqrt-pow1 N/A

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

      13. pow2 N/A

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

      14. sqr-neg-rev N/A

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

      15. lift-neg.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. pow2 N/A

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

      18. sqrt-pow1 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. pow-flip N/A

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

      22. inv-pow N/A

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

      23. metadata-eval N/A

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

      24. lift-neg.f64 N/A

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

      25. frac-2neg N/A

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

      26. clear-num N/A

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

   9. Applied rewrites 23.0%

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

   if 9.99989e-320 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq 10^{-319}:\\ \;\;\;\;\frac{y \cdot x}{\left(-z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x\_m}{z} \leq 5 \cdot 10^{-298}:\\ \;\;\;\;\frac{y \cdot x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* (/ (sin y) y) x_m) z) 5e-298)
    (/ (* y x_m) (* z y))
    (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((((sin(y) / y) * x_m) / z) <= 5e-298) {
		tmp = (y * x_m) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((((sin(y) / y) * x_m) / z) <= 5d-298) then
        tmp = (y * x_m) / (z * y)
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((((Math.sin(y) / y) * x_m) / z) <= 5e-298) {
		tmp = (y * x_m) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if (((math.sin(y) / y) * x_m) / z) <= 5e-298:
		tmp = (y * x_m) / (z * y)
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(sin(y) / y) * x_m) / z) <= 5e-298)
		tmp = Float64(Float64(y * x_m) / Float64(z * y));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((((sin(y) / y) * x_m) / z) <= 5e-298)
		tmp = (y * x_m) / (z * y);
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], 5e-298], N[(N[(y * x$95$m), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 5.0000000000000002e-298

   1. Initial program 97.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 81.5

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

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 47.2

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

   7. Applied rewrites 47.2%

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

   if 5.0000000000000002e-298 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 68.5

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

   5. Applied rewrites 68.5%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq 5 \cdot 10^{-298}:\\ \;\;\;\;\frac{y \cdot x}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 30:\\ \;\;\;\;\frac{x\_m}{z} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot x\_m}{z}\\ \end{array} \end{array} \end{array} \]

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 30.0) (* (/ x_m z) t_0) (/ (* 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 <= 30.0) {
		tmp = (x_m / z) * t_0;
	} 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 <= 30.0d0) then
        tmp = (x_m / z) * t_0
    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 <= 30.0) {
		tmp = (x_m / z) * t_0;
	} 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 <= 30.0:
		tmp = (x_m / z) * t_0
	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 (x_m <= 30.0)
		tmp = Float64(Float64(x_m / z) * t_0);
	else
		tmp = Float64(Float64(t_0 * 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 <= 30.0)
		tmp = (x_m / z) * t_0;
	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[x$95$m, 30.0], N[(N[(x$95$m / z), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(t$95$0 * x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 30

   1. Initial program 97.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* 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 95.5

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

   4. Applied rewrites 95.5%

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

   if 30 < x

   1. Initial program 99.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 30:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y}{y} \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 0.00036:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y} \cdot \frac{\sin y}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 0.00036)
    (* (fma -0.16666666666666666 (* y y) 1.0) (/ x_m z))
    (* (/ x_m y) (/ (sin y) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 0.00036) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * (x_m / z);
	} else {
		tmp = (x_m / y) * (sin(y) / z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 0.00036)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m / y) * Float64(sin(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, 0.00036], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / y), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.60000000000000023e-4

   1. Initial program 97.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* 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 96.5

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

   4. Applied rewrites 96.5%

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

   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 73.2

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

   7. Applied rewrites 73.2%

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

   if 3.60000000000000023e-4 < y

   1. Initial program 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. times-frac N/A

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

      7. div-inv N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. div-inv N/A

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

      13. lower-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 98.3

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

   4. Applied rewrites 98.3%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00036:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{\sin y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.02 \cdot 10^{+148}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z \cdot y} \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.02e+148)
    (* (/ x_m z) (/ (sin y) y))
    (* (/ (sin y) (* z y)) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.02e+148) {
		tmp = (x_m / z) * (sin(y) / y);
	} else {
		tmp = (sin(y) / (z * y)) * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 1.02d+148) then
        tmp = (x_m / z) * (sin(y) / y)
    else
        tmp = (sin(y) / (z * y)) * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.02e+148) {
		tmp = (x_m / z) * (Math.sin(y) / y);
	} else {
		tmp = (Math.sin(y) / (z * y)) * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 1.02e+148:
		tmp = (x_m / z) * (math.sin(y) / y)
	else:
		tmp = (math.sin(y) / (z * y)) * x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 1.02e+148)
		tmp = Float64(Float64(x_m / z) * Float64(sin(y) / y));
	else
		tmp = Float64(Float64(sin(y) / Float64(z * y)) * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 1.02e+148)
		tmp = (x_m / z) * (sin(y) / y);
	else
		tmp = (sin(y) / (z * y)) * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 1.02e+148], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.02e148

   1. Initial program 97.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* 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 95.8

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

   4. Applied rewrites 95.8%

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

   if 1.02e148 < x

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 69.6

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

   4. Applied rewrites 69.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. associate-*l/ N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

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

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

      16. lower-*.f64 N/A

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

      17. lower-neg.f64 92.1

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

   6. Applied rewrites 92.1%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{+148}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{z \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 4.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x\_m}{y \cdot y} \cdot y\right) \cdot \frac{y}{z}\\ \end{array} \end{array} \]

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 (<= y 4.5)
    (/
     (*
      (fma (fma 0.008333333333333333 (* y y) -0.16666666666666666) (* y y) 1.0)
      x_m)
     z)
    (* (* (/ x_m (* y y)) 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 <= 4.5) {
		tmp = (fma(fma(0.008333333333333333, (y * y), -0.16666666666666666), (y * y), 1.0) * x_m) / z;
	} else {
		tmp = ((x_m / (y * y)) * 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 <= 4.5)
		tmp = Float64(Float64(fma(fma(0.008333333333333333, Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * x_m) / z);
	else
		tmp = Float64(Float64(Float64(x_m / Float64(y * y)) * 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, 4.5], N[(N[(N[(N[(0.008333333333333333 * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x$95$m / N[(y * y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.5

   1. Initial program 97.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if 4.5 < y

   1. Initial program 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. times-frac N/A

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

      7. div-inv N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. div-inv N/A

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

      13. lower-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 98.3

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

   4. Applied rewrites 98.3%

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

   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 26.6

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

   7. Applied rewrites 26.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

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

      4. flip-- N/A

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

      5. +-lft-identity N/A

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

      6. associate-/l/ N/A

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

      7. +-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub0-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-lft-identity N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 29.8

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

   9. Applied rewrites 29.8%

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

   11. Applied rewrites 38.1%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y \cdot y} \cdot y\right) \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1350000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot x\_m, -0.16666666666666666, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x\_m}{y \cdot y} \cdot y\right) \cdot \frac{y}{z}\\ \end{array} \end{array} \]

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 (<= y 1350000000.0)
    (/ (fma (* (* y y) x_m) -0.16666666666666666 x_m) z)
    (* (* (/ x_m (* y y)) 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 <= 1350000000.0) {
		tmp = fma(((y * y) * x_m), -0.16666666666666666, x_m) / z;
	} else {
		tmp = ((x_m / (y * y)) * 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 <= 1350000000.0)
		tmp = Float64(fma(Float64(Float64(y * y) * x_m), -0.16666666666666666, x_m) / z);
	else
		tmp = Float64(Float64(Float64(x_m / Float64(y * y)) * 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, 1350000000.0], N[(N[(N[(N[(y * y), $MachinePrecision] * x$95$m), $MachinePrecision] * -0.16666666666666666 + x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x$95$m / N[(y * y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.35e9

   1. Initial program 97.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 88.3

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

   4. Applied rewrites 88.3%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 72.2

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

   7. Applied rewrites 72.2%

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

   if 1.35e9 < y

   1. Initial program 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. times-frac N/A

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

      7. div-inv N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. div-inv N/A

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

      13. lower-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 98.3

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

   4. Applied rewrites 98.3%

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

   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 27.0

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

   7. Applied rewrites 27.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-sub0 N/A

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

      4. flip-- N/A

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

      5. +-lft-identity N/A

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

      6. associate-/l/ N/A

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

      7. +-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub0-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-lft-identity N/A

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

      14. lower-*.f64 N/A

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

      15. lower-neg.f64 30.2

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

   9. Applied rewrites 30.2%

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

   11. Applied rewrites 38.7%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1350000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, -0.16666666666666666, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y \cdot y} \cdot y\right) \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1350000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot x\_m, -0.16666666666666666, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x\_m \cdot x\_m\right) \cdot \frac{-1}{z}}{-x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 1350000000.0)
    (/ (fma (* (* y y) x_m) -0.16666666666666666 x_m) z)
    (/ (* (* x_m x_m) (/ -1.0 z)) (- x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1350000000.0) {
		tmp = fma(((y * y) * x_m), -0.16666666666666666, x_m) / z;
	} else {
		tmp = ((x_m * x_m) * (-1.0 / z)) / -x_m;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 1350000000.0)
		tmp = Float64(fma(Float64(Float64(y * y) * x_m), -0.16666666666666666, x_m) / z);
	else
		tmp = Float64(Float64(Float64(x_m * x_m) * Float64(-1.0 / z)) / Float64(-x_m));
	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, 1350000000.0], N[(N[(N[(N[(y * y), $MachinePrecision] * x$95$m), $MachinePrecision] * -0.16666666666666666 + x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision] / (-x$95$m)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.35e9

   1. Initial program 97.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 88.3

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

   4. Applied rewrites 88.3%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 72.2

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

   7. Applied rewrites 72.2%

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

   if 1.35e9 < y

   1. Initial program 97.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 21.3

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

   5. Applied rewrites 21.3%

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

   7. Applied rewrites 21.3%

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

   9. Applied rewrites 33.8%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1350000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, -0.16666666666666666, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot \frac{-1}{z}}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1350000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot x\_m, -0.16666666666666666, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{y}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 1350000000.0)
    (/ (fma (* (* y y) x_m) -0.16666666666666666 x_m) z)
    (/ (/ y z) (/ y x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1350000000.0) {
		tmp = fma(((y * y) * x_m), -0.16666666666666666, x_m) / z;
	} else {
		tmp = (y / z) / (y / x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 1350000000.0)
		tmp = Float64(fma(Float64(Float64(y * y) * x_m), -0.16666666666666666, x_m) / z);
	else
		tmp = Float64(Float64(y / z) / Float64(y / x_m));
	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, 1350000000.0], N[(N[(N[(N[(y * y), $MachinePrecision] * x$95$m), $MachinePrecision] * -0.16666666666666666 + x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.35e9

   1. Initial program 97.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 88.3

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

   4. Applied rewrites 88.3%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 72.2

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

   7. Applied rewrites 72.2%

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

   if 1.35e9 < y

   1. Initial program 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. times-frac N/A

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

      7. div-inv N/A

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

      8. *-lft-identity N/A

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

      9. associate-*l/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. div-inv N/A

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

      13. lower-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. *-lft-identity N/A

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

      16. lower-/.f64 98.3

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

   4. Applied rewrites 98.3%

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

   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 27.0

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

   7. Applied rewrites 27.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 28.3

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

   9. Applied rewrites 28.3%

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

Alternative 16: 56.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1350000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot x\_m, -0.16666666666666666, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{z \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 1350000000.0)
    (/ (fma (* (* y y) x_m) -0.16666666666666666 x_m) z)
    (/ (* x_m x_m) (* z x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1350000000.0) {
		tmp = fma(((y * y) * x_m), -0.16666666666666666, x_m) / z;
	} else {
		tmp = (x_m * x_m) / (z * x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 1350000000.0)
		tmp = Float64(fma(Float64(Float64(y * y) * x_m), -0.16666666666666666, x_m) / z);
	else
		tmp = Float64(Float64(x_m * x_m) / Float64(z * x_m));
	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, 1350000000.0], N[(N[(N[(N[(y * y), $MachinePrecision] * x$95$m), $MachinePrecision] * -0.16666666666666666 + x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.35e9

   1. Initial program 97.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 88.3

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

   4. Applied rewrites 88.3%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 72.2

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

   7. Applied rewrites 72.2%

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

   if 1.35e9 < y

   1. Initial program 97.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 21.3

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

   5. Applied rewrites 21.3%

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

   7. Applied rewrites 21.3%

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

   9. Applied rewrites 30.9%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1350000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot x, -0.16666666666666666, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1350000000:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{z \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 1350000000.0)
    (* (fma -0.16666666666666666 (* y y) 1.0) (/ x_m z))
    (/ (* x_m x_m) (* z x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 1350000000.0) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * (x_m / z);
	} else {
		tmp = (x_m * x_m) / (z * x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 1350000000.0)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m * x_m) / Float64(z * x_m));
	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, 1350000000.0], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.35e9

   1. Initial program 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* 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 96.5

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

   4. Applied rewrites 96.5%

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

   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 72.7

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

   7. Applied rewrites 72.7%

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

   if 1.35e9 < y

   1. Initial program 97.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 21.3

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

   5. Applied rewrites 21.3%

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

   7. Applied rewrites 21.3%

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

   9. Applied rewrites 30.9%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1350000000:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.0% 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 97.8%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 60.6

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

5. Applied rewrites 60.6%

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

Developer Target 1: 99.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024298 
(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))