Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.0% → 99.7%

Time: 8.5s

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: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.4 \cdot 10^{+23}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot t\_0}{z}\\ \end{array} \end{array} \end{array} \]

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 3.4e+23) (* 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 <= 3.4e+23) {
		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 <= 3.4d+23) 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 <= 3.4e+23) {
		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 <= 3.4e+23:
		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 <= 3.4e+23)
		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 <= 3.4e+23)
		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, 3.4e+23], 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 < 3.39999999999999992e23

   1. Initial program 96.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.7

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

   4. Applied rewrites 95.7%

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

   if 3.39999999999999992e23 < x

   1. Initial program 99.6%

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

Alternative 2: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot t\_0}{z} \leq -1 \cdot 10^{+51}:\\ \;\;\;\;\frac{x\_m}{y \cdot z} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

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+51)
      (* (/ x_m (* y z)) (sin y))
      (* 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+51) {
		tmp = (x_m / (y * z)) * sin(y);
	} 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+51)) then
        tmp = (x_m / (y * z)) * sin(y)
    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+51) {
		tmp = (x_m / (y * z)) * Math.sin(y);
	} 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+51:
		tmp = (x_m / (y * z)) * math.sin(y)
	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+51)
		tmp = Float64(Float64(x_m / Float64(y * z)) * sin(y));
	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+51)
		tmp = (x_m / (y * z)) * sin(y);
	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+51], N[(N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $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) < -1e51

   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. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. distribute-frac-neg2 N/A

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

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

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

      14. remove-double-neg N/A

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

      15. frac-2neg N/A

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

      16. associate-*l/ N/A

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

      17. *-lft-identity N/A

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

      18. associate-/l/ N/A

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

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

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

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

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

      21. lower-/.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-neg.f64 N/A

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

      25. lower-neg.f64 74.1

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

   4. Applied rewrites 74.1%

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

   if -1e51 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 96.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.6

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

   4. Applied rewrites 97.6%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \frac{\sin y}{y}}{z} \leq -1 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999999995:\\ \;\;\;\;\frac{x\_m}{y \cdot z} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) 0.999999995)
    (* (/ x_m (* y z)) (sin y))
    (/ (* x_m (fma (* -0.16666666666666666 y) y 1.0)) 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 tmp;
	if ((sin(y) / y) <= 0.999999995) {
		tmp = (x_m / (y * z)) * sin(y);
	} else {
		tmp = (x_m * fma((-0.16666666666666666 * y), y, 1.0)) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.999999995)
		tmp = Float64(Float64(x_m / Float64(y * z)) * sin(y));
	else
		tmp = Float64(Float64(x_m * fma(Float64(-0.16666666666666666 * y), y, 1.0)) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.999999995], N[(N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99999999500000003

   1. Initial program 94.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. distribute-frac-neg2 N/A

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

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

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

      14. remove-double-neg N/A

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

      15. frac-2neg N/A

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

      16. associate-*l/ N/A

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

      17. *-lft-identity N/A

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

      18. associate-/l/ N/A

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

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

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

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

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

      21. lower-/.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

      24. lower-neg.f64 N/A

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

      25. lower-neg.f64 91.8

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

   4. Applied rewrites 91.8%

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

   if 0.99999999500000003 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}}{z} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999999995:\\ \;\;\;\;\frac{x}{y \cdot z} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999999995:\\ \;\;\;\;x\_m \cdot \frac{\sin y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) 0.999999995)
    (* x_m (/ (sin y) (* z y)))
    (/ (* x_m (fma (* -0.16666666666666666 y) y 1.0)) 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 tmp;
	if ((sin(y) / y) <= 0.999999995) {
		tmp = x_m * (sin(y) / (z * y));
	} else {
		tmp = (x_m * fma((-0.16666666666666666 * y), y, 1.0)) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.999999995)
		tmp = Float64(x_m * Float64(sin(y) / Float64(z * y)));
	else
		tmp = Float64(Float64(x_m * fma(Float64(-0.16666666666666666 * y), y, 1.0)) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.999999995], N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.99999999500000003

   1. Initial program 94.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 91.5

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

   4. Applied rewrites 91.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 91.7

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

   6. Applied rewrites 91.7%

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

   if 0.99999999500000003 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}}{z} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999999995:\\ \;\;\;\;x \cdot \frac{\sin y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.999998:\\ \;\;\;\;\frac{\sin y \cdot x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (sin y) y) 0.999998)
    (/ (* (sin y) x_m) (* z y))
    (/ (* x_m (fma (* -0.16666666666666666 y) y 1.0)) 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 tmp;
	if ((sin(y) / y) <= 0.999998) {
		tmp = (sin(y) * x_m) / (z * y);
	} else {
		tmp = (x_m * fma((-0.16666666666666666 * y), y, 1.0)) / z;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(sin(y) / y) <= 0.999998)
		tmp = Float64(Float64(sin(y) * x_m) / Float64(z * y));
	else
		tmp = Float64(Float64(x_m * fma(Float64(-0.16666666666666666 * y), y, 1.0)) / z);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.999998], N[(N[(N[Sin[y], $MachinePrecision] * x$95$m), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.999998000000000054

   1. Initial program 94.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 91.6

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

   4. Applied rewrites 91.6%

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

   if 0.999998000000000054 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {y}^{2}\right)}}{z} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

Alternative 6: 65.8% accurate, 0.6× speedup

Math

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

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
  (pow (* (fma (/ (* y y) x_m) 0.16666666666666666 (pow x_m -1.0)) z) -1.0)))

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 * pow((fma(((y * y) / x_m), 0.16666666666666666, pow(x_m, -1.0)) * z), -1.0);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * (Float64(fma(Float64(Float64(y * y) / x_m), 0.16666666666666666, (x_m ^ -1.0)) * z) ^ -1.0))
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[Power[N[(N[(N[(N[(y * y), $MachinePrecision] / x$95$m), $MachinePrecision] * 0.16666666666666666 + N[Power[x$95$m, -1.0], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 95.4

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

4. Applied rewrites 95.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. lift-/.f64 N/A

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

   6. frac-times N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-*.f64 N/A

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

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

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

   15. lower-*.f64 N/A

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

   16. lower-neg.f64 89.2

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

6. Applied rewrites 89.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. frac-2neg N/A

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

   9. frac-times N/A

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

   10. associate-/l/ N/A

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

   11. associate-*r/ N/A

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

   12. lift-sin.f64 N/A

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

   13. div-inv N/A

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

   14. lift-sin.f64 N/A

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

   15. associate-*r/ N/A

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

   16. clear-num N/A

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

   17. frac-times N/A

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

   18. metadata-eval N/A

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

   19. lower-/.f64 N/A

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

   20. lower-*.f64 N/A

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

8. Applied rewrites 90.9%

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

9. Taylor expanded in y around 0

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

    1. *-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-/.f64 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 N/A

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

    6. lower-/.f64 66.3

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

11. Applied rewrites 66.3%

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

12. Final simplification 66.3%

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

Alternative 7: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{+92}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{\sin y \cdot x\_m} \cdot z\right)}^{-1}\\ \end{array} \end{array} \]

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 5e+92)
    (* (/ (sin y) y) (/ x_m z))
    (pow (* (/ y (* (sin y) x_m)) z) -1.0))))

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 <= 5e+92) {
		tmp = (sin(y) / y) * (x_m / z);
	} else {
		tmp = pow(((y / (sin(y) * x_m)) * z), -1.0);
	}
	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 <= 5d+92) then
        tmp = (sin(y) / y) * (x_m / z)
    else
        tmp = ((y / (sin(y) * x_m)) * z) ** (-1.0d0)
    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 <= 5e+92) {
		tmp = (Math.sin(y) / y) * (x_m / z);
	} else {
		tmp = Math.pow(((y / (Math.sin(y) * x_m)) * z), -1.0);
	}
	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 <= 5e+92:
		tmp = (math.sin(y) / y) * (x_m / z)
	else:
		tmp = math.pow(((y / (math.sin(y) * x_m)) * z), -1.0)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 5e+92)
		tmp = Float64(Float64(sin(y) / y) * Float64(x_m / z));
	else
		tmp = Float64(Float64(y / Float64(sin(y) * x_m)) * z) ^ -1.0;
	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 <= 5e+92)
		tmp = (sin(y) / y) * (x_m / z);
	else
		tmp = ((y / (sin(y) * x_m)) * z) ^ -1.0;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 5e+92], N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(y / N[(N[Sin[y], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], -1.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000022e92

   1. Initial program 96.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.9

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

   4. Applied rewrites 95.9%

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

   if 5.00000000000000022e92 < x

   1. Initial program 99.6%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. 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 93.0

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

   4. Applied rewrites 93.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 84.2

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

   6. Applied rewrites 84.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. frac-2neg N/A

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

      9. frac-times N/A

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

      10. associate-/l/ N/A

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

      11. associate-*r/ N/A

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

      12. lift-sin.f64 N/A

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

      13. div-inv N/A

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

      14. lift-sin.f64 N/A

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

      15. associate-*r/ N/A

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

      16. clear-num N/A

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

      17. frac-times N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

   8. Applied rewrites 99.6%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+92}:\\ \;\;\;\;\frac{\sin y}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{y}{\sin y \cdot x} \cdot z\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq 4 \cdot 10^{-317}:\\ \;\;\;\;\frac{x\_m \cdot y}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m (/ (sin y) y)) z) 4e-317)
    (/ (* x_m y) (* z y))
    (/ x_m z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (sin(y) / y)) / z) <= 4e-317) {
		tmp = (x_m * y) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x_m * (sin(y) / y)) / z) <= 4d-317) then
        tmp = (x_m * y) / (z * y)
    else
        tmp = x_m / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * (Math.sin(y) / y)) / z) <= 4e-317) {
		tmp = (x_m * y) / (z * y);
	} else {
		tmp = x_m / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if ((x_m * (math.sin(y) / y)) / z) <= 4e-317:
		tmp = (x_m * y) / (z * y)
	else:
		tmp = x_m / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= 4e-317)
		tmp = Float64(Float64(x_m * y) / Float64(z * y));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (((x_m * (sin(y) / y)) / z) <= 4e-317)
		tmp = (x_m * y) / (z * y);
	else
		tmp = x_m / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 4e-317], N[(N[(x$95$m * y), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 3.99999993e-317

   1. Initial program 95.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 88.7

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 51.8

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

   7. Applied rewrites 51.8%

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

   if 3.99999993e-317 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z)

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 60.9

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

   5. Applied rewrites 60.9%

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

Alternative 9: 60.8% accurate, 0.9× speedup

Math

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

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)) 0.0) (/ (* x_m x_m) (* z x_m)) (/ 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)) <= 0.0) {
		tmp = (x_m * x_m) / (z * x_m);
	} 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)) <= 0.0d0) then
        tmp = (x_m * x_m) / (z * x_m)
    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)) <= 0.0) {
		tmp = (x_m * x_m) / (z * x_m);
	} 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)) <= 0.0:
		tmp = (x_m * x_m) / (z * x_m)
	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(x_m * Float64(sin(y) / y)) <= 0.0)
		tmp = Float64(Float64(x_m * x_m) / Float64(z * x_m));
	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)) <= 0.0)
		tmp = (x_m * x_m) / (z * x_m);
	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[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(z * x$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 56.1

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

   5. Applied rewrites 56.1%

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

   7. Applied rewrites 56.2%

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

   9. Applied rewrites 45.7%

      \[\leadsto \frac{1}{\frac{0 \cdot x - \left(-x\right) \cdot \left(-z\right)}{\color{blue}{\left(-x\right) \cdot x}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 45.5%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 56.8

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

   5. Applied rewrites 56.8%

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

Alternative 10: 57.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 6.2:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-1} \cdot \frac{x\_m \cdot x\_m}{z}\\ \end{array} \end{array} \]

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 (<= y 6.2)
    (/ (* x_m (fma (* -0.16666666666666666 y) y 1.0)) z)
    (* (pow x_m -1.0) (/ (* x_m 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 (y <= 6.2) {
		tmp = (x_m * fma((-0.16666666666666666 * y), y, 1.0)) / z;
	} else {
		tmp = pow(x_m, -1.0) * ((x_m * 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 (y <= 6.2)
		tmp = Float64(Float64(x_m * fma(Float64(-0.16666666666666666 * y), y, 1.0)) / z);
	else
		tmp = Float64((x_m ^ -1.0) * Float64(Float64(x_m * 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[y, 6.2], N[(N[(x$95$m * N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[Power[x$95$m, -1.0], $MachinePrecision] * N[(N[(x$95$m * x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.20000000000000018

   1. Initial program 97.1%

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

   7. Applied rewrites 68.9%

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

   if 6.20000000000000018 < y

   1. Initial program 96.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 19.1

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

   5. Applied rewrites 19.1%

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

   7. Applied rewrites 19.3%

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

   9. Applied rewrites 23.9%

      \[\leadsto \frac{1}{\frac{0 \cdot x - \left(-x\right) \cdot \left(-z\right)}{\color{blue}{\left(-x\right) \cdot x}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 27.4%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1} \cdot \frac{x \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 6.2:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;{\left(x\_m \cdot \frac{z}{x\_m \cdot x\_m}\right)}^{-1}\\ \end{array} \end{array} \]

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 6.2)
    (/ (* x_m (fma (* -0.16666666666666666 y) y 1.0)) z)
    (pow (* x_m (/ z (* x_m x_m))) -1.0))))

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 <= 6.2) {
		tmp = (x_m * fma((-0.16666666666666666 * y), y, 1.0)) / z;
	} else {
		tmp = pow((x_m * (z / (x_m * x_m))), -1.0);
	}
	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 <= 6.2)
		tmp = Float64(Float64(x_m * fma(Float64(-0.16666666666666666 * y), y, 1.0)) / z);
	else
		tmp = Float64(x_m * Float64(z / Float64(x_m * x_m))) ^ -1.0;
	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, 6.2], N[(N[(x$95$m * N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[Power[N[(x$95$m * N[(z / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.20000000000000018

   1. Initial program 97.1%

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.9

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

   5. Applied rewrites 68.9%

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

   7. Applied rewrites 68.9%

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

   if 6.20000000000000018 < y

   1. Initial program 96.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 19.1

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

   5. Applied rewrites 19.1%

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

   7. Applied rewrites 19.3%

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

   9. Applied rewrites 23.9%

      \[\leadsto \frac{1}{\frac{0 \cdot x - \left(-x\right) \cdot \left(-z\right)}{\color{blue}{\left(-x\right) \cdot x}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 27.4%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2:\\ \;\;\;\;\frac{x \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot \frac{z}{x \cdot x}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.2% accurate, 1.0× speedup

Math

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

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 (pow (* (fma (* 0.16666666666666666 y) y 1.0) (/ z x_m)) -1.0)))

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 * pow((fma((0.16666666666666666 * y), y, 1.0) * (z / x_m)), -1.0);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * (Float64(fma(Float64(0.16666666666666666 * y), y, 1.0) * Float64(z / x_m)) ^ -1.0))
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[Power[N[(N[(N[(0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 95.4

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

4. Applied rewrites 95.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. lift-/.f64 N/A

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

   6. frac-times N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-neg.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-*.f64 N/A

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

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

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

   15. lower-*.f64 N/A

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

   16. lower-neg.f64 89.2

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

6. Applied rewrites 89.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. frac-2neg N/A

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

   9. frac-times N/A

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

   10. associate-/l/ N/A

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

   11. associate-*r/ N/A

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

   12. lift-sin.f64 N/A

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

   13. div-inv N/A

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

   14. lift-sin.f64 N/A

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

   15. associate-*r/ N/A

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

   16. clear-num N/A

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

   17. frac-times N/A

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

   18. metadata-eval N/A

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

   19. lower-/.f64 N/A

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

   20. lower-*.f64 N/A

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

8. Applied rewrites 90.9%

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

9. Taylor expanded in y around 0

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

    1. associate-/l* N/A

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

    2. associate-*r* N/A

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

    3. distribute-lft1-in N/A

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

    4. lower-*.f64 N/A

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

    5. unpow2 N/A

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

    6. associate-*r* N/A

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

    7. lower-fma.f64 N/A

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

    8. lower-*.f64 N/A

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

    9. lower-/.f64 66.2

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

11. Applied rewrites 66.2%

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

12. Final simplification 66.2%

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

Alternative 13: 56.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{+40}:\\ \;\;\;\;\frac{x\_m \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot y}{z \cdot y}\\ \end{array} \end{array} \]

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 3.4e+40)
    (/ (* x_m (fma (* -0.16666666666666666 y) y 1.0)) z)
    (/ (* x_m y) (* z 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 (z <= 3.4e+40) {
		tmp = (x_m * fma((-0.16666666666666666 * y), y, 1.0)) / z;
	} else {
		tmp = (x_m * y) / (z * 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 (z <= 3.4e+40)
		tmp = Float64(Float64(x_m * fma(Float64(-0.16666666666666666 * y), y, 1.0)) / z);
	else
		tmp = Float64(Float64(x_m * y) / Float64(z * 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[z, 3.4e+40], N[(N[(x$95$m * N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m * y), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.39999999999999989e40

   1. Initial program 96.3%

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 48.7

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

   5. Applied rewrites 48.7%

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

   7. Applied rewrites 48.7%

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

   if 3.39999999999999989e40 < z

   1. Initial program 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 91.3

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

   4. Applied rewrites 91.3%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 77.1

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

   7. Applied rewrites 77.1%

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

Alternative 14: 57.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, y \cdot y, 1\right) \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot y}{z \cdot y}\\ \end{array} \end{array} \]

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 1.75e+94)
    (* (fma -0.16666666666666666 (* y y) 1.0) (/ x_m z))
    (/ (* x_m y) (* z 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 <= 1.75e+94) {
		tmp = fma(-0.16666666666666666, (y * y), 1.0) * (x_m / z);
	} else {
		tmp = (x_m * y) / (z * 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 <= 1.75e+94)
		tmp = Float64(fma(-0.16666666666666666, Float64(y * y), 1.0) * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m * y) / Float64(z * 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, 1.75e+94], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.7499999999999999e94

   1. Initial program 97.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 96.6

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

   4. Applied rewrites 96.6%

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

   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 65.1

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

   7. Applied rewrites 65.1%

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

   if 1.7499999999999999e94 < y

   1. Initial program 95.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 91.7

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in y around 0

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

      1. lower-*.f64 32.0

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

   7. Applied rewrites 32.0%

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

Alternative 15: 58.7% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.0%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 56.4

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

5. Applied rewrites 56.4%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
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 2024322 
(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))