Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.2% → 99.5%

Time: 11.7s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.2% 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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{+81}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot t\_0}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (* x_s (if (<= x_m 2e+81) (* t_0 (/ x_m z)) (/ (* x_m t_0) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (x_m <= 2e+81) {
		tmp = t_0 * (x_m / z);
	} else {
		tmp = (x_m * t_0) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (x_m <= 2d+81) then
        tmp = t_0 * (x_m / z)
    else
        tmp = (x_m * t_0) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (x_m <= 2e+81) {
		tmp = t_0 * (x_m / z);
	} else {
		tmp = (x_m * t_0) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if x_m <= 2e+81:
		tmp = t_0 * (x_m / z)
	else:
		tmp = (x_m * t_0) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (x_m <= 2e+81)
		tmp = Float64(t_0 * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m * t_0) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (x_m <= 2e+81)
		tmp = t_0 * (x_m / z);
	else
		tmp = (x_m * t_0) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 2e+81], N[(t$95$0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.99999999999999984e81

   1. Initial program 92.7%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 96.7

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

   4. Applied rewrites 96.7%

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

   if 1.99999999999999984e81 < x

   1. Initial program 99.7%

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

Alternative 2: 40.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \frac{\sin y}{y}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-260}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot \left(x\_m \cdot \left(y \cdot y\right)\right)}{z}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{1}{y \cdot z} \cdot \left(x\_m \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* x_m (/ (sin y) y)) z)))
   (*
    x_s
    (if (<= t_0 -1e-260)
      (/ (* -0.16666666666666666 (* x_m (* y y))) z)
      (if (<= t_0 0.0) (* (/ 1.0 (* y z)) (* x_m y)) (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -1e-260) {
		tmp = (-0.16666666666666666 * (x_m * (y * y))) / z;
	} else if (t_0 <= 0.0) {
		tmp = (1.0 / (y * z)) * (x_m * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_m * (sin(y) / y)) / z
    if (t_0 <= (-1d-260)) then
        tmp = ((-0.16666666666666666d0) * (x_m * (y * y))) / z
    else if (t_0 <= 0.0d0) then
        tmp = (1.0d0 / (y * z)) * (x_m * y)
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (Math.sin(y) / y)) / z;
	double tmp;
	if (t_0 <= -1e-260) {
		tmp = (-0.16666666666666666 * (x_m * (y * y))) / z;
	} else if (t_0 <= 0.0) {
		tmp = (1.0 / (y * z)) * (x_m * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = (x_m * (math.sin(y) / y)) / z
	tmp = 0
	if t_0 <= -1e-260:
		tmp = (-0.16666666666666666 * (x_m * (y * y))) / z
	elif t_0 <= 0.0:
		tmp = (1.0 / (y * z)) * (x_m * y)
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m * Float64(sin(y) / y)) / z)
	tmp = 0.0
	if (t_0 <= -1e-260)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(x_m * Float64(y * y))) / z);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(1.0 / Float64(y * z)) * Float64(x_m * y));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (x_m * (sin(y) / y)) / z;
	tmp = 0.0;
	if (t_0 <= -1e-260)
		tmp = (-0.16666666666666666 * (x_m * (y * y))) / z;
	elseif (t_0 <= 0.0)
		tmp = (1.0 / (y * z)) * (x_m * y);
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -1e-260], N[(N[(-0.16666666666666666 * N[(x$95$m * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(1.0 / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -9.99999999999999961e-261

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 5.5

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

   8. Applied rewrites 5.5%

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

   if -9.99999999999999961e-261 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -0.0

   1. Initial program 80.5%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. clear-num N/A

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

      5. div-inv N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 72.4

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

   7. Applied rewrites 72.4%

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

   if -0.0 < (/.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 64.0

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

   5. Applied rewrites 64.0%

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

Alternative 3: 93.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot t\_0}{z} \leq -1 \cdot 10^{-136}:\\ \;\;\;\;\frac{x\_m \cdot \sin y}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (sin y) y)))
   (*
    x_s
    (if (<= (/ (* x_m t_0) z) -1e-136)
      (/ (* x_m (sin y)) (* y z))
      (* t_0 (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = sin(y) / y;
	double tmp;
	if (((x_m * t_0) / z) <= -1e-136) {
		tmp = (x_m * sin(y)) / (y * z);
	} else {
		tmp = t_0 * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(y) / y
    if (((x_m * t_0) / z) <= (-1d-136)) then
        tmp = (x_m * sin(y)) / (y * z)
    else
        tmp = t_0 * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = Math.sin(y) / y;
	double tmp;
	if (((x_m * t_0) / z) <= -1e-136) {
		tmp = (x_m * Math.sin(y)) / (y * z);
	} else {
		tmp = t_0 * (x_m / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = math.sin(y) / y
	tmp = 0
	if ((x_m * t_0) / z) <= -1e-136:
		tmp = (x_m * math.sin(y)) / (y * z)
	else:
		tmp = t_0 * (x_m / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(sin(y) / y)
	tmp = 0.0
	if (Float64(Float64(x_m * t_0) / z) <= -1e-136)
		tmp = Float64(Float64(x_m * sin(y)) / Float64(y * z));
	else
		tmp = Float64(t_0 * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = sin(y) / y;
	tmp = 0.0;
	if (((x_m * t_0) / z) <= -1e-136)
		tmp = (x_m * sin(y)) / (y * z);
	else
		tmp = t_0 * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[N[(N[(x$95$m * t$95$0), $MachinePrecision] / z), $MachinePrecision], -1e-136], N[(N[(x$95$m * N[Sin[y], $MachinePrecision]), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -1e-136

   1. Initial program 99.0%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 89.6

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

   4. Applied rewrites 89.6%

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

   if -1e-136 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 91.7%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.2

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

   4. Applied rewrites 97.2%

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

Alternative 4: 55.8% accurate, 0.8× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (/ (sin y) y)) z) 0.0)
    (* (/ 1.0 (* y z)) (* x_m y))
    (/ x_m z))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.4%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. clear-num N/A

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

      5. div-inv N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 91.7

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 59.4

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

   7. Applied rewrites 59.4%

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

   if -0.0 < (/.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 64.0

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

   5. Applied rewrites 64.0%

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

Alternative 5: 54.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 62.4

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

   5. Applied rewrites 62.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 62.3

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

   7. Applied rewrites 62.3%

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

      1. associate-/r/ N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. neg-sub0 N/A

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

      6. flip-- N/A

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

      7. +-lft-identity N/A

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

      8. associate-/r/ N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. metadata-eval N/A

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

      12. sub0-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-*.f64 55.4

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

   9. Applied rewrites 55.4%

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

   if -0.0 < (/.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 64.0

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

   5. Applied rewrites 64.0%

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

4. Final simplification 58.5%

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

Alternative 6: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{-170}:\\ \;\;\;\;x\_m \cdot \frac{\frac{\sin y}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z 1.55e-170)
    (* x_m (/ (/ (sin y) z) y))
    (* (/ (sin y) y) (/ x_m z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= 1.55e-170) {
		tmp = x_m * ((sin(y) / z) / y);
	} else {
		tmp = (sin(y) / y) * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.55d-170) then
        tmp = x_m * ((sin(y) / z) / y)
    else
        tmp = (sin(y) / y) * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= 1.55e-170) {
		tmp = x_m * ((Math.sin(y) / z) / y);
	} else {
		tmp = (Math.sin(y) / y) * (x_m / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if z <= 1.55e-170:
		tmp = x_m * ((math.sin(y) / z) / y)
	else:
		tmp = (math.sin(y) / y) * (x_m / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= 1.55e-170)
		tmp = Float64(x_m * Float64(Float64(sin(y) / z) / y));
	else
		tmp = Float64(Float64(sin(y) / y) * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= 1.55e-170)
		tmp = x_m * ((sin(y) / z) / y);
	else
		tmp = (sin(y) / y) * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, 1.55e-170], N[(x$95$m * N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.54999999999999993e-170

   1. Initial program 92.9%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 90.8

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

   4. Applied rewrites 90.8%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 93.9

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

   6. Applied rewrites 93.9%

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

   if 1.54999999999999993e-170 < z

   1. Initial program 96.1%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. 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 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{-170}:\\ \;\;\;\;x \cdot \frac{\frac{\sin y}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 5.5 \cdot 10^{-201}:\\ \;\;\;\;\frac{\sin y}{z} \cdot \frac{x\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z 5.5e-201)
    (* (/ (sin y) z) (/ x_m y))
    (* (/ (sin y) y) (/ x_m z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= 5.5e-201) {
		tmp = (sin(y) / z) * (x_m / y);
	} else {
		tmp = (sin(y) / y) * (x_m / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 5.5d-201) then
        tmp = (sin(y) / z) * (x_m / y)
    else
        tmp = (sin(y) / y) * (x_m / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= 5.5e-201) {
		tmp = (Math.sin(y) / z) * (x_m / y);
	} else {
		tmp = (Math.sin(y) / y) * (x_m / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if z <= 5.5e-201:
		tmp = (math.sin(y) / z) * (x_m / y)
	else:
		tmp = (math.sin(y) / y) * (x_m / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= 5.5e-201)
		tmp = Float64(Float64(sin(y) / z) * Float64(x_m / y));
	else
		tmp = Float64(Float64(sin(y) / y) * Float64(x_m / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= 5.5e-201)
		tmp = (sin(y) / z) * (x_m / y);
	else
		tmp = (sin(y) / y) * (x_m / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, 5.5e-201], N[(N[(N[Sin[y], $MachinePrecision] / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.50000000000000034e-201

   1. Initial program 93.8%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. *-lft-identity N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 84.9

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

   4. Applied rewrites 84.9%

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

   if 5.50000000000000034e-201 < z

   1. Initial program 94.4%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 98.8

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

   4. Applied rewrites 98.8%

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

Alternative 8: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \sin y}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 2e-7)
    (/ x_m (/ z (fma -0.16666666666666666 (* y y) 1.0)))
    (/ (* x_m (sin y)) (* y z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 2e-7) {
		tmp = x_m / (z / fma(-0.16666666666666666, (y * y), 1.0));
	} else {
		tmp = (x_m * sin(y)) / (y * z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 2e-7)
		tmp = Float64(x_m / Float64(z / fma(-0.16666666666666666, Float64(y * y), 1.0)));
	else
		tmp = Float64(Float64(x_m * sin(y)) / Float64(y * z));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 2e-7], N[(x$95$m / N[(z / N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[Sin[y], $MachinePrecision]), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.9999999999999999e-7

   1. Initial program 96.7%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.4

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

   4. Applied rewrites 97.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-neg.f64 97.1

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

   6. Applied rewrites 97.1%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 70.0

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

   9. Applied rewrites 70.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. flip3-+ N/A

          \[\leadsto \frac{\color{blue}{\frac{{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3} + {1}^{3}}{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) + \left(1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot 1\right)}}}{\mathsf{neg}\left(z\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

       4. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3} + {1}^{3}}{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) + \left(1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot 1\right)}}{\color{blue}{\mathsf{neg}\left(z\right)}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

       5. flip3-+ N/A

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

       6. lift-*.f64 N/A

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

       7. lift-fma.f64 N/A

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

       8. lift-/.f64 N/A

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

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

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

       10. *-commutative N/A

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

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

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

       12. lift-/.f64 N/A

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

       13. lift-neg.f64 N/A

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

       14. distribute-frac-neg2 N/A

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

       15. remove-double-neg N/A

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

   11. Applied rewrites 70.2%

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

   if 1.9999999999999999e-7 < y

   1. Initial program 85.3%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 91.9

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

   4. Applied rewrites 91.9%

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

Alternative 9: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x\_m}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 4.5e-7)
    (/ x_m (/ z (fma -0.16666666666666666 (* y y) 1.0)))
    (* (sin y) (/ x_m (* y z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 4.5e-7) {
		tmp = x_m / (z / fma(-0.16666666666666666, (y * y), 1.0));
	} else {
		tmp = sin(y) * (x_m / (y * z));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 4.5e-7)
		tmp = Float64(x_m / Float64(z / fma(-0.16666666666666666, Float64(y * y), 1.0)));
	else
		tmp = Float64(sin(y) * Float64(x_m / Float64(y * z)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 4.5e-7], N[(x$95$m / N[(z / N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.4999999999999998e-7

   1. Initial program 96.7%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.4

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

   4. Applied rewrites 97.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-neg.f64 97.1

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

   6. Applied rewrites 97.1%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 70.0

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

   9. Applied rewrites 70.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. flip3-+ N/A

          \[\leadsto \frac{\color{blue}{\frac{{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3} + {1}^{3}}{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) + \left(1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot 1\right)}}}{\mathsf{neg}\left(z\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

       4. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3} + {1}^{3}}{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) + \left(1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot 1\right)}}{\color{blue}{\mathsf{neg}\left(z\right)}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

       5. flip3-+ N/A

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

       6. lift-*.f64 N/A

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

       7. lift-fma.f64 N/A

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

       8. lift-/.f64 N/A

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

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

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

       10. *-commutative N/A

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

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

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

       12. lift-/.f64 N/A

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

       13. lift-neg.f64 N/A

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

       14. distribute-frac-neg2 N/A

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

       15. remove-double-neg N/A

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

   11. Applied rewrites 70.2%

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

   if 4.4999999999999998e-7 < y

   1. Initial program 85.3%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. *-lft-identity N/A

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

      7. associate-*l/ N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-lft-identity N/A

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

      14. associate-/l/ N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 91.9

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

   4. Applied rewrites 91.9%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\frac{z}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot z} \cdot \left(x\_m \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 3.7e+15)
    (*
     (/ x_m z)
     (fma
      y
      (*
       y
       (fma
        y
        (* y (fma y (* y -0.0001984126984126984) 0.008333333333333333))
        -0.16666666666666666))
      1.0))
    (* (/ 1.0 (* y z)) (* x_m y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 3.7e+15) {
		tmp = (x_m / z) * fma(y, (y * fma(y, (y * fma(y, (y * -0.0001984126984126984), 0.008333333333333333)), -0.16666666666666666)), 1.0);
	} else {
		tmp = (1.0 / (y * z)) * (x_m * y);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 3.7e+15)
		tmp = Float64(Float64(x_m / z) * fma(y, Float64(y * fma(y, Float64(y * fma(y, Float64(y * -0.0001984126984126984), 0.008333333333333333)), -0.16666666666666666)), 1.0));
	else
		tmp = Float64(Float64(1.0 / Float64(y * z)) * Float64(x_m * y));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 3.7e+15], N[(N[(x$95$m / z), $MachinePrecision] * N[(y * N[(y * N[(y * N[(y * N[(y * N[(y * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.7e15

   1. Initial program 96.9%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)} \cdot \frac{x}{z} \]

   if 3.7e15 < y

   1. Initial program 83.2%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. clear-num N/A

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

      5. div-inv N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 90.8

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

   4. Applied rewrites 90.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 27.8

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

   7. Applied rewrites 27.8%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y, y \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot z} \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot z} \cdot \left(x\_m \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 3.7e+15)
    (/ x_m (/ z (fma -0.16666666666666666 (* y y) 1.0)))
    (* (/ 1.0 (* y z)) (* x_m y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 3.7e+15) {
		tmp = x_m / (z / fma(-0.16666666666666666, (y * y), 1.0));
	} else {
		tmp = (1.0 / (y * z)) * (x_m * y);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 3.7e+15)
		tmp = Float64(x_m / Float64(z / fma(-0.16666666666666666, Float64(y * y), 1.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(y * z)) * Float64(x_m * y));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 3.7e+15], N[(x$95$m / N[(z / N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.7e15

   1. Initial program 96.9%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.5

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

   4. Applied rewrites 97.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-neg.f64 97.2

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

   6. Applied rewrites 97.2%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.4

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

   9. Applied rewrites 69.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. flip3-+ N/A

          \[\leadsto \frac{\color{blue}{\frac{{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3} + {1}^{3}}{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) + \left(1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot 1\right)}}}{\mathsf{neg}\left(z\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

       4. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3} + {1}^{3}}{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) + \left(1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot 1\right)}}{\color{blue}{\mathsf{neg}\left(z\right)}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

       5. flip3-+ N/A

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

       6. lift-*.f64 N/A

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

       7. lift-fma.f64 N/A

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

       8. lift-/.f64 N/A

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

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

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

       10. *-commutative N/A

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

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

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

       12. lift-/.f64 N/A

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

       13. lift-neg.f64 N/A

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

       14. distribute-frac-neg2 N/A

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

       15. remove-double-neg N/A

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

   11. Applied rewrites 69.6%

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

   if 3.7e15 < y

   1. Initial program 83.2%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. clear-num N/A

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

      5. div-inv N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 90.8

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

   4. Applied rewrites 90.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 27.8

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

   7. Applied rewrites 27.8%

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

Alternative 12: 57.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot z} \cdot \left(x\_m \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 3.7e+15)
    (/ (* x_m (fma -0.16666666666666666 (* y y) 1.0)) z)
    (* (/ 1.0 (* y z)) (* x_m y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 3.7e+15) {
		tmp = (x_m * fma(-0.16666666666666666, (y * y), 1.0)) / z;
	} else {
		tmp = (1.0 / (y * z)) * (x_m * y);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 3.7e+15)
		tmp = Float64(Float64(x_m * fma(-0.16666666666666666, Float64(y * y), 1.0)) / z);
	else
		tmp = Float64(Float64(1.0 / Float64(y * z)) * Float64(x_m * y));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 3.7e+15], N[(N[(x$95$m * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(1.0 / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.7e15

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

   if 3.7e15 < y

   1. Initial program 83.2%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. clear-num N/A

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

      5. div-inv N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 90.8

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

   4. Applied rewrites 90.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 27.8

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

   7. Applied rewrites 27.8%

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

Alternative 13: 58.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot z} \cdot \left(x\_m \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 3.7e+15)
    (* (/ x_m z) (fma -0.16666666666666666 (* y y) 1.0))
    (* (/ 1.0 (* y z)) (* x_m y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 3.7e+15) {
		tmp = (x_m / z) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = (1.0 / (y * z)) * (x_m * y);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 3.7e+15)
		tmp = Float64(Float64(x_m / z) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(Float64(1.0 / Float64(y * z)) * Float64(x_m * y));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 3.7e+15], N[(N[(x$95$m / z), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.7e15

   1. Initial program 96.9%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 70.5

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

   7. Applied rewrites 70.5%

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

   if 3.7e15 < y

   1. Initial program 83.2%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. clear-num N/A

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

      5. div-inv N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 90.8

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

   4. Applied rewrites 90.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 27.8

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

   7. Applied rewrites 27.8%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot z} \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;x\_m \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot z} \cdot \left(x\_m \cdot y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 3.7e+15)
    (* x_m (/ (fma -0.16666666666666666 (* y y) 1.0) z))
    (* (/ 1.0 (* y z)) (* x_m y)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 3.7e+15) {
		tmp = x_m * (fma(-0.16666666666666666, (y * y), 1.0) / z);
	} else {
		tmp = (1.0 / (y * z)) * (x_m * y);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 3.7e+15)
		tmp = Float64(x_m * Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) / z));
	else
		tmp = Float64(Float64(1.0 / Float64(y * z)) * Float64(x_m * y));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 3.7e+15], N[(x$95$m * N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.7e15

   1. Initial program 96.9%

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

      1. lift-sin.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. div-inv N/A

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

      5. frac-2neg N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.5

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

   4. Applied rewrites 97.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. frac-2neg N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-neg.f64 97.2

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

   6. Applied rewrites 97.2%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.4

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

   9. Applied rewrites 69.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. flip3-+ N/A

          \[\leadsto \frac{\color{blue}{\frac{{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3} + {1}^{3}}{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) + \left(1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot 1\right)}}}{\mathsf{neg}\left(z\right)} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

       4. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}^{3} + {1}^{3}}{\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) + \left(1 \cdot 1 - \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot 1\right)}}{\color{blue}{\mathsf{neg}\left(z\right)}} \cdot \left(\mathsf{neg}\left(x\right)\right) \]

       5. flip3-+ N/A

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

       6. lift-*.f64 N/A

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

       7. lift-fma.f64 N/A

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

       8. lift-/.f64 N/A

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

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

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

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

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

       11. lift-/.f64 N/A

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

       12. lift-neg.f64 N/A

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

       13. distribute-frac-neg2 N/A

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

       14. remove-double-neg N/A

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

       15. lower-*.f64 N/A

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

   11. Applied rewrites 69.4%

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

   if 3.7e15 < y

   1. Initial program 83.2%

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

      1. lift-sin.f64 N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. clear-num N/A

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

      5. div-inv N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 90.8

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

   4. Applied rewrites 90.8%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 27.8

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

   7. Applied rewrites 27.8%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y \cdot z} \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.6% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 63.0

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

5. Applied rewrites 63.0%

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

Developer Target 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024214 
(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))