Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 95.9% → 99.7%

Time: 8.4s

Alternatives: 12

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: 95.9% 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

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{\frac{z\_m}{\sin y} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y}{y} \cdot x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 5.3e+17)
    (/ x (* (/ z_m (sin y)) y))
    (/ (* (/ (sin y) y) x) z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 5.3e+17) {
		tmp = x / ((z_m / sin(y)) * y);
	} else {
		tmp = ((sin(y) / y) * x) / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 5.3d+17) then
        tmp = x / ((z_m / sin(y)) * y)
    else
        tmp = ((sin(y) / y) * x) / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 5.3e+17) {
		tmp = x / ((z_m / Math.sin(y)) * y);
	} else {
		tmp = ((Math.sin(y) / y) * x) / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 5.3e+17:
		tmp = x / ((z_m / math.sin(y)) * y)
	else:
		tmp = ((math.sin(y) / y) * x) / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 5.3e+17)
		tmp = Float64(x / Float64(Float64(z_m / sin(y)) * y));
	else
		tmp = Float64(Float64(Float64(sin(y) / y) * x) / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 5.3e+17)
		tmp = x / ((z_m / sin(y)) * y);
	else
		tmp = ((sin(y) / y) * x) / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 5.3e+17], N[(x / N[(N[(z$95$m / N[Sin[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.3e17

   1. Initial program 96.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. remove-double-neg N/A

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

      3. lift-*.f64 N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. distribute-frac-neg2 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-/r/ N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 94.2

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

   4. Applied rewrites 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-*.f64 94.2

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

   6. Applied rewrites 94.2%

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

   if 5.3e17 < z

   1. Initial program 99.8%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{\frac{z}{\sin y} \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y}{y} \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\frac{\sin y}{y} \cdot x}{z\_m}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-181}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \frac{x}{z\_m}\\ \mathbf{elif}\;t\_0 \leq 10^{-299}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{\left(-y\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (/ (* (/ (sin y) y) x) z_m)))
   (*
    z_s
    (if (<= t_0 -5e-181)
      (* (fma -0.16666666666666666 (* y y) 1.0) (/ x z_m))
      (if (<= t_0 1e-299) (* (- 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 t_0 = ((sin(y) / y) * x) / z_m;
	double tmp;
	if (t_0 <= -5e-181) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * (x / z_m);
	} else if (t_0 <= 1e-299) {
		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)
	t_0 = Float64(Float64(Float64(sin(y) / y) * x) / z_m)
	tmp = 0.0
	if (t_0 <= -5e-181)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * Float64(x / z_m));
	elseif (t_0 <= 1e-299)
		tmp = Float64(Float64(-y) * Float64(x / Float64(Float64(-y) * z_m)));
	else
		tmp = 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_] := Block[{t$95$0 = N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] / z$95$m), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t$95$0, -5e-181], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-299], N[((-y) * N[(x / N[((-y) * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -5.0000000000000001e-181

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 65.2

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \frac{x \cdot {y}^{2}}{z} + \frac{x}{z}} \]
   7. 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. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 65.6

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

   8. Applied rewrites 65.6%

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

   if -5.0000000000000001e-181 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 9.99999999999999992e-300

   1. Initial program 91.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.4

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

   7. Applied rewrites 78.4%

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

   if 9.99999999999999992e-300 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 60.8

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

   5. Applied rewrites 60.8%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq -5 \cdot 10^{-181}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq 10^{-299}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{\left(-y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\frac{\sin y}{y} \cdot x}{z\_m}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-181}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \frac{x}{z\_m}\\ \mathbf{elif}\;t\_0 \leq 10^{-299}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{\left(-y\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (/ (* (/ (sin y) y) x) z_m)))
   (*
    z_s
    (if (<= t_0 -5e-181)
      (* (* (* y y) -0.16666666666666666) (/ x z_m))
      (if (<= t_0 1e-299) (* (- 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 t_0 = ((sin(y) / y) * x) / z_m;
	double tmp;
	if (t_0 <= -5e-181) {
		tmp = ((y * y) * -0.16666666666666666) * (x / z_m);
	} else if (t_0 <= 1e-299) {
		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) :: t_0
    real(8) :: tmp
    t_0 = ((sin(y) / y) * x) / z_m
    if (t_0 <= (-5d-181)) then
        tmp = ((y * y) * (-0.16666666666666666d0)) * (x / z_m)
    else if (t_0 <= 1d-299) 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 t_0 = ((Math.sin(y) / y) * x) / z_m;
	double tmp;
	if (t_0 <= -5e-181) {
		tmp = ((y * y) * -0.16666666666666666) * (x / z_m);
	} else if (t_0 <= 1e-299) {
		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):
	t_0 = ((math.sin(y) / y) * x) / z_m
	tmp = 0
	if t_0 <= -5e-181:
		tmp = ((y * y) * -0.16666666666666666) * (x / z_m)
	elif t_0 <= 1e-299:
		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)
	t_0 = Float64(Float64(Float64(sin(y) / y) * x) / z_m)
	tmp = 0.0
	if (t_0 <= -5e-181)
		tmp = Float64(Float64(Float64(y * y) * -0.16666666666666666) * Float64(x / z_m));
	elseif (t_0 <= 1e-299)
		tmp = Float64(Float64(-y) * Float64(x / Float64(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)
	t_0 = ((sin(y) / y) * x) / z_m;
	tmp = 0.0;
	if (t_0 <= -5e-181)
		tmp = ((y * y) * -0.16666666666666666) * (x / z_m);
	elseif (t_0 <= 1e-299)
		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_] := Block[{t$95$0 = N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] / z$95$m), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t$95$0, -5e-181], N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-299], N[((-y) * N[(x / N[((-y) * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -5.0000000000000001e-181

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 65.2

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \frac{x \cdot {y}^{2}}{z} + \frac{x}{z}} \]
   7. 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. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 65.6

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

   8. Applied rewrites 65.6%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 3.7%

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

   if -5.0000000000000001e-181 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 9.99999999999999992e-300

   1. Initial program 91.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 95.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 78.4

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

   7. Applied rewrites 78.4%

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

   if 9.99999999999999992e-300 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 60.8

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

   5. Applied rewrites 60.8%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq -5 \cdot 10^{-181}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq 10^{-299}:\\ \;\;\;\;\left(-y\right) \cdot \frac{x}{\left(-y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999999999998:\\ \;\;\;\;\frac{x}{y} \cdot \frac{\sin y}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (sin y) y) 0.999999999998)
    (* (/ x y) (/ (sin y) z_m))
    (/ x z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 0.999999999998) {
		tmp = (x / y) * (sin(y) / z_m);
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((sin(y) / y) <= 0.999999999998d0) then
        tmp = (x / y) * (sin(y) / z_m)
    else
        tmp = x / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((Math.sin(y) / y) <= 0.999999999998) {
		tmp = (x / y) * (Math.sin(y) / z_m);
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if (math.sin(y) / y) <= 0.999999999998:
		tmp = (x / y) * (math.sin(y) / z_m)
	else:
		tmp = x / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.999999999998)
		tmp = Float64(Float64(x / y) * Float64(sin(y) / z_m));
	else
		tmp = Float64(x / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if ((sin(y) / y) <= 0.999999999998)
		tmp = (x / y) * (sin(y) / z_m);
	else
		tmp = x / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.999999999998], N[(N[(x / y), $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99999999999800004

   1. Initial program 93.6%

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 91.7

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

   4. Applied rewrites 91.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-*l/ N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 93.7

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

   6. Applied rewrites 93.7%

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

   if 0.99999999999800004 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999999999998:\\ \;\;\;\;\frac{x}{y} \cdot \frac{\sin y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999999999998:\\ \;\;\;\;\frac{\sin y}{y \cdot z\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (sin y) y) 0.999999999998)
    (* (/ (sin y) (* y z_m)) x)
    (/ 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) <= 0.999999999998) {
		tmp = (sin(y) / (y * z_m)) * x;
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((sin(y) / y) <= 0.999999999998d0) then
        tmp = (sin(y) / (y * z_m)) * x
    else
        tmp = x / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((Math.sin(y) / y) <= 0.999999999998) {
		tmp = (Math.sin(y) / (y * z_m)) * x;
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if (math.sin(y) / y) <= 0.999999999998:
		tmp = (math.sin(y) / (y * z_m)) * x
	else:
		tmp = x / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.999999999998)
		tmp = Float64(Float64(sin(y) / Float64(y * z_m)) * x);
	else
		tmp = Float64(x / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if ((sin(y) / y) <= 0.999999999998)
		tmp = (sin(y) / (y * z_m)) * x;
	else
		tmp = x / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.999999999998], N[(N[(N[Sin[y], $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99999999999800004

   1. Initial program 93.6%

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 91.7

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

   4. Applied rewrites 91.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. remove-double-neg N/A

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

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

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

      6. lift-neg.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

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

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

      12. remove-double-neg N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 91.1

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

   6. Applied rewrites 91.1%

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

   if 0.99999999999800004 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999999999998:\\ \;\;\;\;\frac{\sin y}{y \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999999999998:\\ \;\;\;\;\frac{x}{y \cdot z\_m} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (sin y) y) 0.999999999998)
    (* (/ x (* y z_m)) (sin y))
    (/ x z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((sin(y) / y) <= 0.999999999998) {
		tmp = (x / (y * z_m)) * sin(y);
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((sin(y) / y) <= 0.999999999998d0) then
        tmp = (x / (y * z_m)) * sin(y)
    else
        tmp = x / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((Math.sin(y) / y) <= 0.999999999998) {
		tmp = (x / (y * z_m)) * Math.sin(y);
	} else {
		tmp = x / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if (math.sin(y) / y) <= 0.999999999998:
		tmp = (x / (y * z_m)) * math.sin(y)
	else:
		tmp = x / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.999999999998)
		tmp = Float64(Float64(x / Float64(y * z_m)) * sin(y));
	else
		tmp = Float64(x / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if ((sin(y) / y) <= 0.999999999998)
		tmp = (x / (y * z_m)) * sin(y);
	else
		tmp = x / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.999999999998], N[(N[(x / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(x / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99999999999800004

   1. Initial program 93.6%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. remove-double-neg N/A

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

      3. lift-*.f64 N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. distribute-frac-neg2 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-/r/ N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 91.0

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

   4. Applied rewrites 91.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/r/ N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

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

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

      9. frac-2neg N/A

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

      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. *-commutative N/A

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

      13. lower-*.f64 91.0

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

   6. Applied rewrites 91.0%

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

   if 0.99999999999800004 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 7: 52.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z\_m} \leq -2 \cdot 10^{-50}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \cdot \frac{x}{z\_m}\\ \end{array} \end{array} \]

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) x) z_m) -2e-50)
    (* (* (* y y) -0.16666666666666666) (/ x z_m))
    (* (/ 1.0 (fma 0.16666666666666666 (* y y) 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) * x) / z_m) <= -2e-50) {
		tmp = ((y * y) * -0.16666666666666666) * (x / z_m);
	} else {
		tmp = (1.0 / fma(0.16666666666666666, (y * y), 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(Float64(Float64(sin(y) / y) * x) / z_m) <= -2e-50)
		tmp = Float64(Float64(Float64(y * y) * -0.16666666666666666) * Float64(x / z_m));
	else
		tmp = Float64(Float64(1.0 / fma(0.16666666666666666, Float64(y * y), 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[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] / z$95$m), $MachinePrecision], -2e-50], N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -2.00000000000000002e-50

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \frac{x \cdot {y}^{2}}{z} + \frac{x}{z}} \]
   7. 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. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 65.7

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

   8. Applied rewrites 65.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 3.8%

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

   if -2.00000000000000002e-50 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 96.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 57.8

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \frac{x \cdot {y}^{2}}{z} + \frac{x}{z}} \]
   7. 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. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 54.4

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

   8. Applied rewrites 54.4%

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

   10. Applied rewrites 54.4%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 69.2%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq -2 \cdot 10^{-50}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.16666666666666666, y \cdot y, 1\right)} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z\_m} \leq -2 \cdot 10^{-50}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot z\_m, 0.16666666666666666, z\_m\right)}\\ \end{array} \end{array} \]

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) -2e-50)
    (* (* (* y y) -0.16666666666666666) (/ x z_m))
    (/ 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) {
	double tmp;
	if ((((sin(y) / y) * x) / z_m) <= -2e-50) {
		tmp = ((y * y) * -0.16666666666666666) * (x / z_m);
	} else {
		tmp = x / fma(((y * y) * z_m), 0.16666666666666666, 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) <= -2e-50)
		tmp = Float64(Float64(Float64(y * y) * -0.16666666666666666) * Float64(x / z_m));
	else
		tmp = Float64(x / fma(Float64(Float64(y * y) * z_m), 0.16666666666666666, 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[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] / z$95$m), $MachinePrecision], -2e-50], N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[(y * y), $MachinePrecision] * z$95$m), $MachinePrecision] * 0.16666666666666666 + z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -2.00000000000000002e-50

   1. Initial program 99.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 65.0

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

   5. Applied rewrites 65.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot \frac{x \cdot {y}^{2}}{z} + \frac{x}{z}} \]
   7. 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. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 65.7

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

   8. Applied rewrites 65.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 3.8%

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

   if -2.00000000000000002e-50 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 96.0%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. remove-double-neg N/A

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

      3. lift-*.f64 N/A

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

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

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

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

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. distribute-frac-neg2 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-/r/ N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-neg.f64 87.5

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

   4. Applied rewrites 87.5%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. distribute-frac-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-*.f64 87.5

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

   6. Applied rewrites 87.5%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 68.3

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

   9. Applied rewrites 68.3%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq -2 \cdot 10^{-50}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot -0.16666666666666666\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\left(y \cdot y\right) \cdot z, 0.16666666666666666, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.3 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{\sin y}{y}}{z\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z\_m}}{y} \cdot \sin y\\ \end{array} \end{array} \]

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 (<= z_m 1.3e+112)
    (* (/ (/ (sin y) y) z_m) x)
    (* (/ (/ x z_m) y) (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 (z_m <= 1.3e+112) {
		tmp = ((sin(y) / y) / z_m) * x;
	} else {
		tmp = ((x / z_m) / y) * 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 (z_m <= 1.3d+112) then
        tmp = ((sin(y) / y) / z_m) * x
    else
        tmp = ((x / z_m) / y) * 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 (z_m <= 1.3e+112) {
		tmp = ((Math.sin(y) / y) / z_m) * x;
	} else {
		tmp = ((x / z_m) / y) * 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 z_m <= 1.3e+112:
		tmp = ((math.sin(y) / y) / z_m) * x
	else:
		tmp = ((x / z_m) / y) * 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 (z_m <= 1.3e+112)
		tmp = Float64(Float64(Float64(sin(y) / y) / z_m) * x);
	else
		tmp = Float64(Float64(Float64(x / z_m) / y) * 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 (z_m <= 1.3e+112)
		tmp = ((sin(y) / y) / z_m) * x;
	else
		tmp = ((x / z_m) / y) * 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[z$95$m, 1.3e+112], N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] / z$95$m), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(x / z$95$m), $MachinePrecision] / y), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.3e112

   1. Initial program 96.3%

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 98.0

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

   4. Applied rewrites 98.0%

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

   if 1.3e112 < z

   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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. div-inv N/A

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

      11. associate-*l/ N/A

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

      12. *-lft-identity N/A

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

      13. associate-*l/ N/A

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

      14. div-inv N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

Alternative 10: 62.8% accurate, 4.0× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

2. if y < 2e-8

   1. Initial program 98.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 74.5

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

   5. Applied rewrites 74.5%

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

   if 2e-8 < y

   1. Initial program 92.4%

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

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 40.3

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

   7. Applied rewrites 40.3%

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

4. Final simplification 65.0%

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

Alternative 11: 59.4% accurate, 5.6× speedup

Math

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

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 (/ z_m x))))

C

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

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 * (1.0d0 / (z_m / x))
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 * (1.0 / (z_m / x));
}

Python

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

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(1.0 / Float64(z_m / x)))
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 * (1.0 / (z_m / x));
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[(1.0 / N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.8%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 59.5

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

5. Applied rewrites 59.5%

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

7. Applied rewrites 59.5%

   \[\leadsto \frac{1}{\color{blue}{\frac{z}{x}}} \]
8. Add Preprocessing

Alternative 12: 59.5% 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 96.8%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 59.5

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

5. Applied rewrites 59.5%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
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 2024276 
(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))