Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.1% → 99.7%

Time: 9.2s

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: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\sin y}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{+42}:\\ \;\;\;\;\frac{x\_m}{z} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot x\_m}{z}\\ \end{array} \end{array} \end{array} \]

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 1e+42) (* (/ x_m z) t_0) (/ (* 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 <= 1e+42) {
		tmp = (x_m / z) * t_0;
	} 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 <= 1d+42) then
        tmp = (x_m / z) * t_0
    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 <= 1e+42) {
		tmp = (x_m / z) * t_0;
	} 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 <= 1e+42:
		tmp = (x_m / z) * t_0
	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 (x_m <= 1e+42)
		tmp = Float64(Float64(x_m / z) * t_0);
	else
		tmp = Float64(Float64(t_0 * 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 <= 1e+42)
		tmp = (x_m / z) * t_0;
	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[x$95$m, 1e+42], N[(N[(x$95$m / z), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(t$95$0 * x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000004e42

   1. Initial program 92.9%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. 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 92.8

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

   4. Applied rewrites 92.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-sin.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lower-/.f64 97.5

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

   6. Applied rewrites 97.5%

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

   if 1.00000000000000004e42 < x

   1. Initial program 99.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+42}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin y}{y} \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 50.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\frac{\sin y}{y} \cdot x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-116}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{x\_m}{\left(-y\right) \cdot z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* (/ (sin y) y) x_m) z)))
   (*
    x_s
    (if (<= t_0 -2e-116)
      (* (* -0.16666666666666666 (* y y)) (/ x_m z))
      (if (<= t_0 0.0) (* (/ 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 t_0 = ((sin(y) / y) * x_m) / z;
	double tmp;
	if (t_0 <= -2e-116) {
		tmp = (-0.16666666666666666 * (y * y)) * (x_m / z);
	} else if (t_0 <= 0.0) {
		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) :: t_0
    real(8) :: tmp
    t_0 = ((sin(y) / y) * x_m) / z
    if (t_0 <= (-2d-116)) then
        tmp = ((-0.16666666666666666d0) * (y * y)) * (x_m / z)
    else if (t_0 <= 0.0d0) 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 t_0 = ((Math.sin(y) / y) * x_m) / z;
	double tmp;
	if (t_0 <= -2e-116) {
		tmp = (-0.16666666666666666 * (y * y)) * (x_m / z);
	} else if (t_0 <= 0.0) {
		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):
	t_0 = ((math.sin(y) / y) * x_m) / z
	tmp = 0
	if t_0 <= -2e-116:
		tmp = (-0.16666666666666666 * (y * y)) * (x_m / z)
	elif t_0 <= 0.0:
		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)
	t_0 = Float64(Float64(Float64(sin(y) / y) * x_m) / z)
	tmp = 0.0
	if (t_0 <= -2e-116)
		tmp = Float64(Float64(-0.16666666666666666 * Float64(y * y)) * Float64(x_m / z));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x_m / Float64(Float64(-y) * z)) * Float64(-y));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = ((sin(y) / y) * x_m) / z;
	tmp = 0.0;
	if (t_0 <= -2e-116)
		tmp = (-0.16666666666666666 * (y * y)) * (x_m / z);
	elseif (t_0 <= 0.0)
		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_] := Block[{t$95$0 = N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -2e-116], N[(N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(x$95$m / N[((-y) * z), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -2e-116

   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. 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 86.4

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

   4. Applied rewrites 86.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-sin.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lower-/.f64 94.9

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

   6. Applied rewrites 94.9%

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

   7. Taylor expanded in y around 0

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 75.6

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

   9. Applied rewrites 75.6%

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

   10. Taylor expanded in y around inf

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

   12. Applied rewrites 7.3%

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

   if -2e-116 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 0.0

   1. Initial program 84.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 67.4

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

   7. Applied rewrites 67.4%

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

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

   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 61.7

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

   5. Applied rewrites 61.7%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq -2 \cdot 10^{-116}:\\ \;\;\;\;\left(-0.16666666666666666 \cdot \left(y \cdot y\right)\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq 0:\\ \;\;\;\;\frac{x}{\left(-y\right) \cdot z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{\frac{\sin y}{y} \cdot x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-116}:\\ \;\;\;\;\left(\frac{y \cdot y}{z} \cdot -0.16666666666666666\right) \cdot x\_m\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{x\_m}{\left(-y\right) \cdot z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* (/ (sin y) y) x_m) z)))
   (*
    x_s
    (if (<= t_0 -2e-116)
      (* (* (/ (* y y) z) -0.16666666666666666) x_m)
      (if (<= t_0 0.0) (* (/ 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 t_0 = ((sin(y) / y) * x_m) / z;
	double tmp;
	if (t_0 <= -2e-116) {
		tmp = (((y * y) / z) * -0.16666666666666666) * x_m;
	} else if (t_0 <= 0.0) {
		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) :: t_0
    real(8) :: tmp
    t_0 = ((sin(y) / y) * x_m) / z
    if (t_0 <= (-2d-116)) then
        tmp = (((y * y) / z) * (-0.16666666666666666d0)) * x_m
    else if (t_0 <= 0.0d0) 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 t_0 = ((Math.sin(y) / y) * x_m) / z;
	double tmp;
	if (t_0 <= -2e-116) {
		tmp = (((y * y) / z) * -0.16666666666666666) * x_m;
	} else if (t_0 <= 0.0) {
		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):
	t_0 = ((math.sin(y) / y) * x_m) / z
	tmp = 0
	if t_0 <= -2e-116:
		tmp = (((y * y) / z) * -0.16666666666666666) * x_m
	elif t_0 <= 0.0:
		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)
	t_0 = Float64(Float64(Float64(sin(y) / y) * x_m) / z)
	tmp = 0.0
	if (t_0 <= -2e-116)
		tmp = Float64(Float64(Float64(Float64(y * y) / z) * -0.16666666666666666) * x_m);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x_m / Float64(Float64(-y) * z)) * Float64(-y));
	else
		tmp = Float64(x_m / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = ((sin(y) / y) * x_m) / z;
	tmp = 0.0;
	if (t_0 <= -2e-116)
		tmp = (((y * y) / z) * -0.16666666666666666) * x_m;
	elseif (t_0 <= 0.0)
		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_] := Block[{t$95$0 = N[(N[(N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -2e-116], N[(N[(N[(N[(y * y), $MachinePrecision] / z), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(x$95$m / N[((-y) * z), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < -2e-116

   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 80.6

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

   4. Applied rewrites 80.6%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. associate-*r/ N/A

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

      11. associate-*l/ N/A

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

      12. associate-*r/ N/A

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

      13. distribute-lft1-in N/A

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

      14. +-commutative N/A

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

      15. associate-*r/ N/A

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

      16. *-rgt-identity N/A

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

      17. lower-/.f64 N/A

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

   7. Applied rewrites 75.4%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 7.5%

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

   if -2e-116 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 0.0

   1. Initial program 84.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 67.4

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

   7. Applied rewrites 67.4%

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

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

   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 61.7

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

   5. Applied rewrites 61.7%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq -2 \cdot 10^{-116}:\\ \;\;\;\;\left(\frac{y \cdot y}{z} \cdot -0.16666666666666666\right) \cdot x\\ \mathbf{elif}\;\frac{\frac{\sin y}{y} \cdot x}{z} \leq 0:\\ \;\;\;\;\frac{x}{\left(-y\right) \cdot z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.9999030129349976:\\ \;\;\;\;\frac{\sin y \cdot x\_m}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z}\\ \end{array} \end{array} \]

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 (<= (/ (sin y) y) 0.9999030129349976)
    (/ (* (sin y) x_m) (* 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 ((sin(y) / y) <= 0.9999030129349976) {
		tmp = (sin(y) * x_m) / (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 ((sin(y) / y) <= 0.9999030129349976d0) then
        tmp = (sin(y) * x_m) / (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 ((Math.sin(y) / y) <= 0.9999030129349976) {
		tmp = (Math.sin(y) * x_m) / (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 (math.sin(y) / y) <= 0.9999030129349976:
		tmp = (math.sin(y) * x_m) / (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(sin(y) / y) <= 0.9999030129349976)
		tmp = Float64(Float64(sin(y) * x_m) / 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 ((sin(y) / y) <= 0.9999030129349976)
		tmp = (sin(y) * x_m) / (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[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.9999030129349976], N[(N[(N[Sin[y], $MachinePrecision] * x$95$m), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 0.999903012934997615

   1. Initial program 88.6%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 88.3

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

   4. Applied rewrites 88.3%

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

   if 0.999903012934997615 < (/.f64 (sin.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 5: 95.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.0001:\\ \;\;\;\;\frac{x\_m}{z \cdot y} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot x\_m}{z}\\ \end{array} \end{array} \]

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 (<= (/ (sin y) y) 0.0001)
    (* (/ x_m (* z y)) (sin y))
    (/
     (*
      (fma
       (fma
        (fma -0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        -0.16666666666666666)
       (* y y)
       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) <= 0.0001) {
		tmp = (x_m / (z * y)) * sin(y);
	} else {
		tmp = (fma(fma(fma(-0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), -0.16666666666666666), (y * y), 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) <= 0.0001)
		tmp = Float64(Float64(x_m / Float64(z * y)) * sin(y));
	else
		tmp = Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * 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], 0.0001], N[(N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 y) y) < 1.00000000000000005e-4

   1. Initial program 88.5%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. 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 93.3

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

   4. Applied rewrites 93.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-sin.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lower-/.f64 94.3

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

   6. Applied rewrites 94.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. frac-2neg N/A

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

      10. lift-neg.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. associate-/r* N/A

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

      13. lift-neg.f64 N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg N/A

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

      16. lower-/.f64 N/A

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

      17. clear-num N/A

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

      18. frac-2neg N/A

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

      19. metadata-eval N/A

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

      20. lift-neg.f64 N/A

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

      21. associate-/r/ N/A

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

      22. metadata-eval N/A

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

      23. mul-1-neg N/A

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

      24. lift-neg.f64 N/A

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

      25. remove-double-neg N/A

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

      26. lower-*.f64 88.2

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

   8. Applied rewrites 88.2%

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

   if 1.00000000000000005e-4 < (/.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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right)}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin y}{y} \leq 0.0001:\\ \;\;\;\;\frac{x}{z \cdot y} \cdot \sin y\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.2% accurate, 0.8× speedup

Math

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

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

   1. Initial program 90.4%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. 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 87.7

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 57.1

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

   7. Applied rewrites 57.1%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 61.7

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

   5. Applied rewrites 61.7%

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

4. Final simplification 59.1%

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

Alternative 7: 95.9% accurate, 1.0× speedup

Math

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

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) (/ (sin y) y))))

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) * (sin(y) / y));
}

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) * (sin(y) / y))
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) * (Math.sin(y) / y));
}

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) * (math.sin(y) / y))

Julia

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

Derivation

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. 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 91.8

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

4. Applied rewrites 91.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-sin.f64 N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. lift-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lower-/.f64 97.3

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

6. Applied rewrites 97.3%

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

7. Final simplification 97.3%

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

Alternative 8: 59.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 13500:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(\frac{\frac{x\_m}{z}}{y \cdot y} \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 13500.0)
    (/
     (*
      (fma
       (fma
        (fma -0.0001984126984126984 (* y y) 0.008333333333333333)
        (* y y)
        -0.16666666666666666)
       (* y y)
       1.0)
      x_m)
     z)
    (* (- y) (* (/ (/ x_m z) (* y y)) (- 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 <= 13500.0) {
		tmp = (fma(fma(fma(-0.0001984126984126984, (y * y), 0.008333333333333333), (y * y), -0.16666666666666666), (y * y), 1.0) * x_m) / z;
	} else {
		tmp = -y * (((x_m / z) / (y * y)) * -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 <= 13500.0)
		tmp = Float64(Float64(fma(fma(fma(-0.0001984126984126984, Float64(y * y), 0.008333333333333333), Float64(y * y), -0.16666666666666666), Float64(y * y), 1.0) * x_m) / z);
	else
		tmp = Float64(Float64(-y) * Float64(Float64(Float64(x_m / z) / Float64(y * y)) * Float64(-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, 13500.0], N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(y * y), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(y * y), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(y * y), $MachinePrecision] + 1.0), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], N[((-y) * N[(N[(N[(x$95$m / z), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 13500

   1. Initial program 97.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if 13500 < y

   1. Initial program 81.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 30.5

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

   7. Applied rewrites 30.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-sub0 N/A

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

      7. flip-- N/A

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

      8. +-lft-identity N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-*.f64 N/A

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

      14. sub0-neg N/A

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

      15. lower-neg.f64 32.3

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

   9. Applied rewrites 32.3%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 13500:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, y \cdot y, 0.008333333333333333\right), y \cdot y, -0.16666666666666666\right), y \cdot y, 1\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(\frac{\frac{x}{z}}{y \cdot y} \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 13500:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right) \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(\frac{\frac{x\_m}{z}}{y \cdot y} \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 13500.0)
    (* (fma (* -0.16666666666666666 y) y 1.0) (/ x_m z))
    (* (- y) (* (/ (/ x_m z) (* y y)) (- 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 <= 13500.0) {
		tmp = fma((-0.16666666666666666 * y), y, 1.0) * (x_m / z);
	} else {
		tmp = -y * (((x_m / z) / (y * y)) * -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 <= 13500.0)
		tmp = Float64(fma(Float64(-0.16666666666666666 * y), y, 1.0) * Float64(x_m / z));
	else
		tmp = Float64(Float64(-y) * Float64(Float64(Float64(x_m / z) / Float64(y * y)) * Float64(-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, 13500.0], N[(N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[((-y) * N[(N[(N[(x$95$m / z), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 13500

   1. Initial program 97.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. 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 91.4

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

   4. Applied rewrites 91.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-sin.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lower-/.f64 98.1

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

   6. Applied rewrites 98.1%

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

   7. Taylor expanded in y around 0

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.4

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

   9. Applied rewrites 70.4%

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

   11. Applied rewrites 70.4%

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

   if 13500 < y

   1. Initial program 81.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 30.5

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

   7. Applied rewrites 30.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. neg-sub0 N/A

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

      7. flip-- N/A

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

      8. +-lft-identity N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-*.f64 N/A

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

      14. sub0-neg N/A

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

      15. lower-neg.f64 32.3

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

   9. Applied rewrites 32.3%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 13500:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(\frac{\frac{x}{z}}{y \cdot y} \cdot \left(-y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 13500:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right) \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{-y}{x\_m} \cdot z}\\ \end{array} \end{array} \]

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 13500.0)
    (* (fma (* -0.16666666666666666 y) y 1.0) (/ x_m z))
    (/ (- y) (* (/ (- y) x_m) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= 13500.0) {
		tmp = fma((-0.16666666666666666 * y), y, 1.0) * (x_m / z);
	} else {
		tmp = -y / ((-y / x_m) * z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= 13500.0)
		tmp = Float64(fma(Float64(-0.16666666666666666 * y), y, 1.0) * Float64(x_m / z));
	else
		tmp = Float64(Float64(-y) / Float64(Float64(Float64(-y) / 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, 13500.0], N[(N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[((-y) / N[(N[((-y) / x$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 13500

   1. Initial program 97.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. 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 91.4

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

   4. Applied rewrites 91.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-sin.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lower-/.f64 98.1

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

   6. Applied rewrites 98.1%

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

   7. Taylor expanded in y around 0

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.4

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

   9. Applied rewrites 70.4%

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

   11. Applied rewrites 70.4%

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

   if 13500 < y

   1. Initial program 81.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 30.5

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

   7. Applied rewrites 30.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. frac-2neg N/A

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

      7. distribute-frac-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

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

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

      13. *-commutative N/A

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

      14. remove-double-neg N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 31.9

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

   9. Applied rewrites 31.9%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 13500:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{-y}{x} \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 13500:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right) \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x\_m}{y}}{z} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 13500.0)
    (* (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 <= 13500.0) {
		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 <= 13500.0)
		tmp = Float64(fma(Float64(-0.16666666666666666 * y), y, 1.0) * Float64(x_m / z));
	else
		tmp = Float64(Float64(Float64(Float64(-x_m) / y) / z) * Float64(-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, 13500.0], N[(N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-x$95$m) / y), $MachinePrecision] / z), $MachinePrecision] * (-y)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 13500

   1. Initial program 97.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. 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 91.4

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

   4. Applied rewrites 91.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-sin.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lower-/.f64 98.1

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

   6. Applied rewrites 98.1%

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

   7. Taylor expanded in y around 0

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.4

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

   9. Applied rewrites 70.4%

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

   11. Applied rewrites 70.4%

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

   if 13500 < y

   1. Initial program 81.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 30.5

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

   7. Applied rewrites 30.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-frac-neg N/A

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

      8. lift-neg.f64 N/A

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

      9. lower-/.f64 30.8

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

   9. Applied rewrites 30.8%

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

Alternative 12: 60.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 13500:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right) \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(-y\right) \cdot z} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 13500.0)
    (* (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 <= 13500.0) {
		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 <= 13500.0)
		tmp = Float64(fma(Float64(-0.16666666666666666 * y), y, 1.0) * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m / Float64(Float64(-y) * z)) * Float64(-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, 13500.0], N[(N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[((-y) * z), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 13500

   1. Initial program 97.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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. 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 91.4

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

   4. Applied rewrites 91.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-sin.f64 N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lower-/.f64 98.1

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

   6. Applied rewrites 98.1%

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

   7. Taylor expanded in y around 0

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

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.4

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

   9. Applied rewrites 70.4%

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

   11. Applied rewrites 70.4%

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

   if 13500 < y

   1. Initial program 81.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 30.5

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

   7. Applied rewrites 30.5%

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

Alternative 13: 59.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 13500:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)}{z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(-y\right) \cdot z} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= y 13500.0)
    (* (/ (fma (* -0.16666666666666666 y) y 1.0) z) x_m)
    (* (/ 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 <= 13500.0) {
		tmp = (fma((-0.16666666666666666 * y), y, 1.0) / z) * x_m;
	} 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 <= 13500.0)
		tmp = Float64(Float64(fma(Float64(-0.16666666666666666 * y), y, 1.0) / z) * x_m);
	else
		tmp = Float64(Float64(x_m / Float64(Float64(-y) * z)) * Float64(-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, 13500.0], N[(N[(N[(N[(-0.16666666666666666 * y), $MachinePrecision] * y + 1.0), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(x$95$m / N[((-y) * z), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 13500

   1. Initial program 97.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. associate-*r/ N/A

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

      11. associate-*l/ N/A

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

      12. associate-*r/ N/A

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

      13. distribute-lft1-in N/A

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

      14. +-commutative N/A

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

      15. associate-*r/ N/A

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

      16. *-rgt-identity N/A

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

      17. lower-/.f64 N/A

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

   7. Applied rewrites 68.0%

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

   9. Applied rewrites 68.0%

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

   if 13500 < y

   1. Initial program 81.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 30.5

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

   7. Applied rewrites 30.5%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 13500:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666 \cdot y, y, 1\right)}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(-y\right) \cdot z} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(-y\right) \cdot z} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= y 6.5e+68) (/ 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 <= 6.5e+68) {
		tmp = x_m / z;
	} else {
		tmp = (x_m / (-y * z)) * -y;
	}
	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 <= 6.5d+68) then
        tmp = x_m / z
    else
        tmp = (x_m / (-y * z)) * -y
    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 <= 6.5e+68) {
		tmp = x_m / z;
	} else {
		tmp = (x_m / (-y * z)) * -y;
	}
	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 <= 6.5e+68:
		tmp = 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 <= 6.5e+68)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(Float64(x_m / Float64(Float64(-y) * z)) * Float64(-y));
	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 <= 6.5e+68)
		tmp = x_m / z;
	else
		tmp = (x_m / (-y * z)) * -y;
	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, 6.5e+68], N[(x$95$m / z), $MachinePrecision], N[(N[(x$95$m / N[((-y) * z), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.5000000000000005e68

   1. Initial program 97.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 70.1

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

   5. Applied rewrites 70.1%

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

   if 6.5000000000000005e68 < y

   1. Initial program 76.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 32.6

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

   7. Applied rewrites 32.6%

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

Alternative 15: 58.8% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 62.1

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

5. Applied rewrites 62.1%

   \[\leadsto \color{blue}{\frac{x}{z}} \]
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 2024243 
(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))