Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 95.9% → 99.6%

Time: 10.8s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.2 \cdot 10^{-61}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot t\_0}{z}\\ \end{array} \end{array} \end{array} \]

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 3.2e-61) (* 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 <= 3.2e-61) {
		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 <= 3.2d-61) 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 <= 3.2e-61) {
		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 <= 3.2e-61:
		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 <= 3.2e-61)
		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 <= 3.2e-61)
		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, 3.2e-61], 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 < 3.2000000000000001e-61

   1. Initial program 93.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 98.3

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

   4. Applied rewrites 98.3%

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

   if 3.2000000000000001e-61 < 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: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot t\_0}{z} \leq -2 \cdot 10^{+38}:\\ \;\;\;\;\sin y \cdot \frac{x\_m}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

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) -2e+38)
      (* (sin y) (/ x_m (* 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) <= -2e+38) {
		tmp = sin(y) * (x_m / (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) <= (-2d+38)) then
        tmp = sin(y) * (x_m / (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) <= -2e+38) {
		tmp = Math.sin(y) * (x_m / (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) <= -2e+38:
		tmp = math.sin(y) * (x_m / (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) <= -2e+38)
		tmp = Float64(sin(y) * Float64(x_m / 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) <= -2e+38)
		tmp = sin(y) * (x_m / (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], -2e+38], N[(N[Sin[y], $MachinePrecision] * N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -1.99999999999999995e38

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. associate-/l/ N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 83.0

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

   4. Applied rewrites 83.0%

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

   if -1.99999999999999995e38 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.7

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

   4. Applied rewrites 96.7%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -2 \cdot 10^{+38}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq -1 \cdot 10^{-271}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x\_m}{z}}}\\ \end{array} \end{array} \]

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) -1e-271)
    (* (/ x_m z) (fma -0.16666666666666666 (* y y) 1.0))
    (/ 1.0 (/ 1.0 (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (sin(y) / y)) / z) <= -1e-271) {
		tmp = (x_m / z) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = 1.0 / (1.0 / (x_m / z));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= -1e-271)
		tmp = Float64(Float64(x_m / z) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(x_m / z)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], -1e-271], N[(N[(x$95$m / z), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -9.99999999999999963e-272

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.7

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

   4. Applied rewrites 93.7%

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

   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 57.8

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

   7. Applied rewrites 57.8%

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

   if -9.99999999999999963e-272 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 93.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 55.7

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 55.9%

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

   9. Applied rewrites 56.0%

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{z}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -1 \cdot 10^{-271}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{z}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{y \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x\_m}{z}}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) -2.5e-104)
    (/ (fma -0.16666666666666666 (* y (* y y)) y) (* y (/ z x_m)))
    (/ 1.0 (/ 1.0 (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= -2.5e-104) {
		tmp = fma(-0.16666666666666666, (y * (y * y)), y) / (y * (z / x_m));
	} else {
		tmp = 1.0 / (1.0 / (x_m / z));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= -2.5e-104)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * Float64(y * y)), y) / Float64(y * Float64(z / x_m)));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(x_m / z)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], -2.5e-104], N[(N[(-0.16666666666666666 * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] / N[(y * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < -2.49999999999999989e-104

   1. Initial program 91.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. remove-double-div N/A

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

      9. div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. div-inv N/A

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

      12. remove-double-div N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 33.1

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

   7. Applied rewrites 33.1%

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

   if -2.49999999999999989e-104 < (/.f64 (sin.f64 y) y)

   1. Initial program 96.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 61.3

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

   5. Applied rewrites 61.3%

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

   7. Applied rewrites 61.5%

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

   9. Applied rewrites 61.5%

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{z}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666, y \cdot \left(y \cdot y\right), y\right)}{y \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{z}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq -1 \cdot 10^{-271}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (/ (sin y) y)) z) -1e-271)
    (* (/ x_m z) (fma -0.16666666666666666 (* y y) 1.0))
    (/ 1.0 (/ z x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (sin(y) / y)) / z) <= -1e-271) {
		tmp = (x_m / z) * fma(-0.16666666666666666, (y * y), 1.0);
	} else {
		tmp = 1.0 / (z / x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= -1e-271)
		tmp = Float64(Float64(x_m / z) * fma(-0.16666666666666666, Float64(y * y), 1.0));
	else
		tmp = Float64(1.0 / Float64(z / x_m));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], -1e-271], N[(N[(x$95$m / z), $MachinePrecision] * N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -9.99999999999999963e-272

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.7

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

   4. Applied rewrites 93.7%

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

   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 57.8

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

   7. Applied rewrites 57.8%

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

   if -9.99999999999999963e-272 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 93.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 55.7

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 55.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{z}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -1 \cdot 10^{-271}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 37.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq -1 \cdot 10^{-271}:\\ \;\;\;\;x\_m \cdot \frac{\left(y \cdot y\right) \cdot -0.16666666666666666}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (/ (sin y) y)) z) -1e-271)
    (* x_m (/ (* (* y y) -0.16666666666666666) z))
    (/ 1.0 (/ z x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (sin(y) / y)) / z) <= -1e-271) {
		tmp = x_m * (((y * y) * -0.16666666666666666) / z);
	} else {
		tmp = 1.0 / (z / x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x_m * (sin(y) / y)) / z) <= (-1d-271)) then
        tmp = x_m * (((y * y) * (-0.16666666666666666d0)) / z)
    else
        tmp = 1.0d0 / (z / x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (Math.sin(y) / y)) / z) <= -1e-271) {
		tmp = x_m * (((y * y) * -0.16666666666666666) / z);
	} else {
		tmp = 1.0 / (z / x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if ((x_m * (math.sin(y) / y)) / z) <= -1e-271:
		tmp = x_m * (((y * y) * -0.16666666666666666) / z)
	else:
		tmp = 1.0 / (z / x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= -1e-271)
		tmp = Float64(x_m * Float64(Float64(Float64(y * y) * -0.16666666666666666) / z));
	else
		tmp = Float64(1.0 / Float64(z / x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (((x_m * (sin(y) / y)) / z) <= -1e-271)
		tmp = x_m * (((y * y) * -0.16666666666666666) / z);
	else
		tmp = 1.0 / (z / x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], -1e-271], N[(x$95$m * N[(N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -9.99999999999999963e-272

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 86.5

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

   4. Applied rewrites 86.5%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. +-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 57.6

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

   7. Applied rewrites 57.6%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 5.1%

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

   if -9.99999999999999963e-272 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 93.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 55.7

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

   5. Applied rewrites 55.7%

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

   7. Applied rewrites 55.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{z}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -1 \cdot 10^{-271}:\\ \;\;\;\;x \cdot \frac{\left(y \cdot y\right) \cdot -0.16666666666666666}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{x\_m}{y \cdot z} \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot -0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x\_m}{z}}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) -2.5e-104)
    (* (/ x_m (* y z)) (fma y (* (* y y) -0.16666666666666666) y))
    (/ 1.0 (/ 1.0 (/ x_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((sin(y) / y) <= -2.5e-104) {
		tmp = (x_m / (y * z)) * fma(y, ((y * y) * -0.16666666666666666), y);
	} else {
		tmp = 1.0 / (1.0 / (x_m / z));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= -2.5e-104)
		tmp = Float64(Float64(x_m / Float64(y * z)) * fma(y, Float64(Float64(y * y) * -0.16666666666666666), y));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(x_m / z)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], -2.5e-104], N[(N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(y * N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < -2.49999999999999989e-104

   1. Initial program 91.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. associate-/l/ N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 93.4

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

   4. Applied rewrites 93.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 33.1

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

   7. Applied rewrites 33.1%

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

   if -2.49999999999999989e-104 < (/.f64 (sin.f64 y) y)

   1. Initial program 96.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 61.3

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

   5. Applied rewrites 61.3%

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

   7. Applied rewrites 61.5%

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

   9. Applied rewrites 61.5%

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{x}{z}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq -2.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \mathsf{fma}\left(y, \left(y \cdot y\right) \cdot -0.16666666666666666, y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{z}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x\_m}{y \cdot z}\\ \end{array} \end{array} \]

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 3.7e-8) (/ x_m z) (* (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 <= 3.7e-8) {
		tmp = x_m / z;
	} else {
		tmp = sin(y) * (x_m / (y * z));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.7d-8) then
        tmp = x_m / z
    else
        tmp = sin(y) * (x_m / (y * z))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 3.7e-8) {
		tmp = x_m / z;
	} else {
		tmp = Math.sin(y) * (x_m / (y * z));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= 3.7e-8:
		tmp = x_m / z
	else:
		tmp = math.sin(y) * (x_m / (y * z))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 3.7e-8)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(sin(y) * Float64(x_m / Float64(y * z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= 3.7e-8)
		tmp = x_m / z;
	else
		tmp = sin(y) * (x_m / (y * z));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, 3.7e-8], N[(x$95$m / z), $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 < 3.7e-8

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if 3.7e-8 < y

   1. Initial program 88.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. associate-/l/ N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 92.7

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

   4. Applied rewrites 92.7%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+72}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot {\left(z \cdot z\right)}^{-0.5}\\ \end{array} \end{array} \]

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 7e+72)
    (*
     (/ x_m z)
     (fma
      (* y y)
      (fma
       y
       (* y (fma (* y y) -0.0001984126984126984 0.008333333333333333))
       -0.16666666666666666)
      1.0))
    (* x_m (pow (* z z) -0.5)))))

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 <= 7e+72) {
		tmp = (x_m / z) * fma((y * y), fma(y, (y * fma((y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), 1.0);
	} else {
		tmp = x_m * pow((z * z), -0.5);
	}
	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 <= 7e+72)
		tmp = Float64(Float64(x_m / z) * fma(Float64(y * y), fma(y, Float64(y * fma(Float64(y * y), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), 1.0));
	else
		tmp = Float64(x_m * (Float64(z * z) ^ -0.5));
	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, 7e+72], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[Power[N[(z * z), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 7.0000000000000002e72

   1. Initial program 97.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 61.8

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

   7. Applied rewrites 61.8%

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

   if 7.0000000000000002e72 < y

   1. Initial program 89.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 90.7

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 15.5

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

   7. Applied rewrites 15.5%

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

   9. Applied rewrites 25.1%

      \[\leadsto {\left(z \cdot z\right)}^{\color{blue}{-0.5}} \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+72}:\\ \;\;\;\;\frac{x}{z} \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(y, y \cdot \mathsf{fma}\left(y \cdot y, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot {\left(z \cdot z\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.5% accurate, 5.6× speedup

Math

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

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 (/ 1.0 (/ z x_m))))

C

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

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 * (1.0d0 / (z / x_m))
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 * (1.0 / (z / x_m));
}

Python

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

Julia

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

Derivation

1. Initial program 95.6%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 55.7

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

5. Applied rewrites 55.7%

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

7. Applied rewrites 55.8%

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

Alternative 11: 57.6% 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 95.6%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 55.7

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

5. Applied rewrites 55.7%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
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 2024216 
(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))