Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.1% → 99.7%

Time: 5.5s

Alternatives: 11

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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{\frac{y}{\sin y} \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{\sin y}{y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 4e-46) (/ x (* (/ y (sin y)) z_m)) (* (/ x z_m) (/ (sin y) y)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 4e-46) {
		tmp = x / ((y / sin(y)) * z_m);
	} else {
		tmp = (x / z_m) * (sin(y) / y);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 4d-46) then
        tmp = x / ((y / sin(y)) * z_m)
    else
        tmp = (x / z_m) * (sin(y) / y)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 4e-46) {
		tmp = x / ((y / Math.sin(y)) * z_m);
	} else {
		tmp = (x / z_m) * (Math.sin(y) / y);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 4e-46:
		tmp = x / ((y / math.sin(y)) * z_m)
	else:
		tmp = (x / z_m) * (math.sin(y) / y)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 4e-46)
		tmp = Float64(x / Float64(Float64(y / sin(y)) * z_m));
	else
		tmp = Float64(Float64(x / z_m) * Float64(sin(y) / y));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 4e-46)
		tmp = x / ((y / sin(y)) * z_m);
	else
		tmp = (x / z_m) * (sin(y) / y);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 4e-46], N[(x / N[(N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.00000000000000009e-46

   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

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 99.3

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

   4. Applied rewrites 99.3%

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

   if 4.00000000000000009e-46 < z

   1. Initial program 99.4%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. *-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 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{\frac{y}{\sin y} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.99999998:\\ \;\;\;\;\frac{\sin y}{y \cdot z\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot \frac{x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (sin y) y) 0.99999998)
    (* (/ (sin y) (* y z_m)) x)
    (* (fma (* y y) -0.16666666666666666 1.0) (/ x z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 0.99999998) {
		tmp = (sin(y) / (y * z_m)) * x;
	} else {
		tmp = fma((y * y), -0.16666666666666666, 1.0) * (x / z_m);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.99999998)
		tmp = Float64(Float64(sin(y) / Float64(y * z_m)) * x);
	else
		tmp = Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * Float64(x / z_m));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.99999998], N[(N[(N[Sin[y], $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.999999980000000011

   1. Initial program 93.0%

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

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

      4. associate-/r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 94.8

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

   4. Applied rewrites 94.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. frac-times N/A

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

      8. clear-num N/A

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

      9. associate-*l/ N/A

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

      10. frac-2neg N/A

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

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

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

      12. neg-mul-1 N/A

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

      13. times-frac N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. /-rgt-identity N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 94.8

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

   6. Applied rewrites 94.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 94.8

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

   8. Applied rewrites 94.8%

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

   if 0.999999980000000011 < (/.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. associate-*l/ N/A

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

      4. associate-/r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. frac-times N/A

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

      8. clear-num N/A

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

      9. associate-*l/ N/A

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

      10. frac-2neg N/A

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

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

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

      12. neg-mul-1 N/A

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

      13. times-frac N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. /-rgt-identity N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 81.8

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

   6. Applied rewrites 81.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 87.0

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

   8. Applied rewrites 87.0%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*r* N/A

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

       4. distribute-lft1-in N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.99999998:\\ \;\;\;\;\frac{\sin y}{y \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.6% accurate, 0.8× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (* (/ (sin y) y) x) z_m) 0.0) (/ (* y x) (* y z_m)) (/ x z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((((sin(y) / y) * x) / z_m) <= 0.0) {
		tmp = (y * x) / (y * z_m);
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((((sin(y) / y) * x) / z_m) <= 0.0d0) then
        tmp = (y * x) / (y * z_m)
    else
        tmp = x / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((((Math.sin(y) / y) * x) / z_m) <= 0.0) {
		tmp = (y * x) / (y * z_m);
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if (((math.sin(y) / y) * x) / z_m) <= 0.0:
		tmp = (y * x) / (y * z_m)
	else:
		tmp = x / z_m
	return z_s * tmp

Julia

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

MATLAB

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

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] / z$95$m), $MachinePrecision], 0.0], N[(N[(y * x), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.2%

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

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 89.6

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

   4. Applied rewrites 89.6%

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

   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 50.7

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

   7. Applied rewrites 50.7%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 66.4

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

   5. Applied rewrites 66.4%

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

4. Final simplification 56.7%

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

Alternative 4: 66.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 10^{-54}:\\ \;\;\;\;\frac{x}{\left(0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (sin y) y) 1e-54)
    (/ x (* (* 0.16666666666666666 (* y y)) z_m))
    (/ x z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 1e-54) {
		tmp = x / ((0.16666666666666666 * (y * y)) * z_m);
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((sin(y) / y) <= 1d-54) then
        tmp = x / ((0.16666666666666666d0 * (y * y)) * z_m)
    else
        tmp = x / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((Math.sin(y) / y) <= 1e-54) {
		tmp = x / ((0.16666666666666666 * (y * y)) * z_m);
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if (math.sin(y) / y) <= 1e-54:
		tmp = x / ((0.16666666666666666 * (y * y)) * z_m)
	else:
		tmp = x / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 1e-54)
		tmp = Float64(x / Float64(Float64(0.16666666666666666 * Float64(y * y)) * z_m));
	else
		tmp = Float64(x / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if ((sin(y) / y) <= 1e-54)
		tmp = x / ((0.16666666666666666 * (y * y)) * z_m);
	else
		tmp = x / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 1e-54], N[(x / N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 1e-54

   1. Initial program 92.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 94.2

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 33.3

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

   7. Applied rewrites 33.3%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 33.3%

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

   if 1e-54 < (/.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 91.9

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

   5. Applied rewrites 91.9%

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

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{\sin y}{z\_m}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{\sin y}{y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 2.8e-50)
    (* (/ (/ (sin y) z_m) y) x)
    (* (/ x z_m) (/ (sin y) y)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 2.8e-50) {
		tmp = ((sin(y) / z_m) / y) * x;
	} else {
		tmp = (x / z_m) * (sin(y) / y);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 2.8d-50) then
        tmp = ((sin(y) / z_m) / y) * x
    else
        tmp = (x / z_m) * (sin(y) / y)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 2.8e-50) {
		tmp = ((Math.sin(y) / z_m) / y) * x;
	} else {
		tmp = (x / z_m) * (Math.sin(y) / y);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 2.8e-50:
		tmp = ((math.sin(y) / z_m) / y) * x
	else:
		tmp = (x / z_m) * (math.sin(y) / y)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 2.8e-50)
		tmp = Float64(Float64(Float64(sin(y) / z_m) / y) * x);
	else
		tmp = Float64(Float64(x / z_m) * Float64(sin(y) / y));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 2.8e-50)
		tmp = ((sin(y) / z_m) / y) * x;
	else
		tmp = (x / z_m) * (sin(y) / y);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 2.8e-50], N[(N[(N[(N[Sin[y], $MachinePrecision] / z$95$m), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.7999999999999998e-50

   1. Initial program 95.1%

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

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

      4. associate-/r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. frac-times N/A

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

      8. clear-num N/A

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

      9. associate-*l/ N/A

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

      10. frac-2neg N/A

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

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

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

      12. neg-mul-1 N/A

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

      13. times-frac N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. /-rgt-identity N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 93.1

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

   6. Applied rewrites 93.1%

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

   if 2.7999999999999998e-50 < z

   1. Initial program 99.4%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. *-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 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{\sin y}{z}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1e-8) {
		tmp = x / z_m;
	} else {
		tmp = (x / (y * z_m)) * sin(y);
	}
	return z_s * tmp;
}

Fortran

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

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1e-8) {
		tmp = x / z_m;
	} else {
		tmp = (x / (y * z_m)) * Math.sin(y);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 1e-8:
		tmp = x / z_m
	else:
		tmp = (x / (y * z_m)) * math.sin(y)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 1e-8)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(Float64(x / Float64(y * z_m)) * sin(y));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 1e-8)
		tmp = x / z_m;
	else
		tmp = (x / (y * z_m)) * sin(y);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1e-8

   1. Initial program 96.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if 1e-8 < y

   1. Initial program 95.4%

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

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

      4. associate-/r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 94.3

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

   4. Applied rewrites 94.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. frac-times N/A

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

      8. clear-num N/A

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

      9. associate-*l/ N/A

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

      10. frac-2neg N/A

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

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

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

      12. neg-mul-1 N/A

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

      13. times-frac N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. /-rgt-identity N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 94.2

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

   6. Applied rewrites 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 94.4

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

   8. Applied rewrites 94.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. clear-num N/A

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

      5. associate-*r/ N/A

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

      6. div-inv N/A

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

      7. times-frac N/A

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

      8. clear-num N/A

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

      9. /-rgt-identity N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-sin.f64 94.4

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

   10. Applied rewrites 94.4%

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

4. Final simplification 78.7%

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

Alternative 7: 66.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(Float64(Float64(-(-1.0)) / fma(Float64(y * y), 0.16666666666666666, 1.0)) * Float64(x / z_m)))
end

Wolfram

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

Derivation

1. Initial program 96.4%

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

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

   4. associate-/r/ N/A

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

   5. lower-/.f64 N/A

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

   6. div-inv N/A

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 97.3

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 65.0

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

7. Applied rewrites 65.0%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. neg-mul-1 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. times-frac N/A

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

   9. un-div-inv N/A

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

   10. metadata-eval N/A

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

   11. frac-2neg N/A

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

   12. lift-/.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

   16. *-commutative N/A

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

   17. neg-mul-1 N/A

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

   18. lift-neg.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. lower-/.f64 65.3

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

9. Applied rewrites 65.3%

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

10. Final simplification 65.3%

    \[\leadsto \frac{--1}{\mathsf{fma}\left(y \cdot y, 0.16666666666666666, 1\right)} \cdot \frac{x}{z} \]
11. Add Preprocessing

Alternative 8: 61.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 26000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, -0.16666666666666666, 1\right) \cdot \frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 26000.0)
    (* (fma (* y y) -0.16666666666666666 1.0) (/ x z_m))
    (/ x (* (* 0.16666666666666666 (* y y)) z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 26000.0) {
		tmp = fma((y * y), -0.16666666666666666, 1.0) * (x / z_m);
	} else {
		tmp = x / ((0.16666666666666666 * (y * y)) * z_m);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 26000.0)
		tmp = Float64(fma(Float64(y * y), -0.16666666666666666, 1.0) * Float64(x / z_m));
	else
		tmp = Float64(x / Float64(Float64(0.16666666666666666 * Float64(y * y)) * z_m));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 26000.0], N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 26000

   1. Initial program 96.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. frac-times N/A

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

      8. clear-num N/A

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

      9. associate-*l/ N/A

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

      10. frac-2neg N/A

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

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

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

      12. neg-mul-1 N/A

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

      13. times-frac N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. /-rgt-identity N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 86.7

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

   6. Applied rewrites 86.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 90.0

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

   8. Applied rewrites 90.0%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*r* N/A

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

       4. distribute-lft1-in N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 69.1

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

   11. Applied rewrites 69.1%

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

   if 26000 < y

   1. Initial program 95.1%

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

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

      4. associate-/r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 94.0

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

   4. Applied rewrites 94.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 34.0

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

   7. Applied rewrites 34.0%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 34.0%

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

Alternative 9: 66.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	return z_s * ((x / fma((0.16666666666666666 * y), y, 1.0)) / z_m);
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(Float64(x / fma(Float64(0.16666666666666666 * y), y, 1.0)) / z_m))
end

Wolfram

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

Derivation

1. Initial program 96.4%

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

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

   4. associate-/r/ N/A

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

   5. lower-/.f64 N/A

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

   6. div-inv N/A

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 97.3

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 65.0

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

7. Applied rewrites 65.0%

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

9. Applied rewrites 65.0%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/r* N/A

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

    4. lower-/.f64 N/A

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

    5. lower-/.f64 65.0

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

11. Applied rewrites 65.0%

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

Alternative 10: 66.6% accurate, 4.6× speedup

Math

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

FPCore

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

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	return z_s * (x / fma(((y * y) * z_m), 0.16666666666666666, z_m));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(x / fma(Float64(Float64(y * y) * z_m), 0.16666666666666666, z_m)))
end

Wolfram

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

Derivation

1. Initial program 96.4%

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

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

   4. associate-/r/ N/A

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

   5. lower-/.f64 N/A

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

   6. div-inv N/A

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 97.3

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

4. Applied rewrites 97.3%

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

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 65.0

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

7. Applied rewrites 65.0%

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

Alternative 11: 58.6% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 59.1

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

5. Applied rewrites 59.1%

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

Developer Target 1: 99.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024332 
(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))