Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.2% → 99.4%

Time: 11.3s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.75 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{\sin y}{z\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z\_m}}{\frac{y}{\sin y}}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 2.75e-83)
    (/ x (/ y (/ (sin y) z_m)))
    (/ (/ x z_m) (/ y (sin y))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 2.75e-83) {
		tmp = x / (y / (sin(y) / z_m));
	} else {
		tmp = (x / z_m) / (y / sin(y));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 2.75d-83) then
        tmp = x / (y / (sin(y) / z_m))
    else
        tmp = (x / z_m) / (y / sin(y))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 2.75e-83) {
		tmp = x / (y / (Math.sin(y) / z_m));
	} else {
		tmp = (x / z_m) / (y / Math.sin(y));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 2.75e-83:
		tmp = x / (y / (math.sin(y) / z_m))
	else:
		tmp = (x / z_m) / (y / math.sin(y))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 2.75e-83)
		tmp = Float64(x / Float64(y / Float64(sin(y) / z_m)));
	else
		tmp = Float64(Float64(x / z_m) / Float64(y / sin(y)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 2.75e-83)
		tmp = x / (y / (sin(y) / z_m));
	else
		tmp = (x / z_m) / (y / sin(y));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 2.75e-83], N[(x / N[(y / N[(N[Sin[y], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.74999999999999982e-83

   1. Initial program 92.8%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sin y}{y}\right), \color{blue}{\left(\frac{z}{x}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\sin y, y\right), \left(\frac{\color{blue}{z}}{x}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right), \left(\frac{z}{x}\right)\right) \]

      7. /-lowering-/.f64 95.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right), \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

   4. Applied egg-rr 95.7%

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

      1. associate-/r/ N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(z \cdot \frac{y}{\sin y}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{y}{\sin y} \cdot \color{blue}{z}\right)\right) \]

      8. remove-double-div N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{y}{\sin y} \cdot \frac{1}{\color{blue}{\frac{1}{z}}}\right)\right) \]

      9. frac-times N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{y \cdot 1}{\color{blue}{\sin y \cdot \frac{1}{z}}}\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{y}{\color{blue}{\sin y} \cdot \frac{1}{z}}\right)\right) \]

      11. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{y}{\frac{\sin y}{\color{blue}{z}}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{\sin y}{z}\right)}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\sin y, \color{blue}{z}\right)\right)\right) \]

      14. sin-lowering-sin.f64 89.4%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), z\right)\right)\right) \]

   6. Applied egg-rr 89.4%

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

   if 2.74999999999999982e-83 < z

   1. Initial program 98.6%

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

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{\left(\frac{y}{\sin y}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{\color{blue}{y}}{\sin y}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \color{blue}{\sin y}\right)\right) \]

      7. sin-lowering-sin.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \mathsf{sin.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 93.1%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sin y}{y}\right), \color{blue}{\left(\frac{z}{x}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\sin y, y\right), \left(\frac{\color{blue}{z}}{x}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right), \left(\frac{z}{x}\right)\right) \]

      7. /-lowering-/.f64 96.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right), \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

   4. Applied egg-rr 96.2%

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

      1. associate-/r/ N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(z \cdot \frac{y}{\sin y}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{y}{\sin y} \cdot \color{blue}{z}\right)\right) \]

      8. remove-double-div N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{y}{\sin y} \cdot \frac{1}{\color{blue}{\frac{1}{z}}}\right)\right) \]

      9. frac-times N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{y \cdot 1}{\color{blue}{\sin y \cdot \frac{1}{z}}}\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{y}{\color{blue}{\sin y} \cdot \frac{1}{z}}\right)\right) \]

      11. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{y}{\frac{\sin y}{\color{blue}{z}}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{\sin y}{z}\right)}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\sin y, \color{blue}{z}\right)\right)\right) \]

      14. sin-lowering-sin.f64 90.6%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), z\right)\right)\right) \]

   6. Applied egg-rr 90.6%

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

   if 1 < z

   1. Initial program 99.7%

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

Alternative 3: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 0.0055:\\ \;\;\;\;\frac{x + x \cdot \left(\left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z\_m \cdot y}{\sin y}}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 0.0055)
    (/
     (+
      x
      (*
       x
       (* (* y y) (+ -0.16666666666666666 (* (* y y) 0.008333333333333333)))))
     z_m)
    (/ x (/ (* z_m y) (sin y))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 0.0055) {
		tmp = (x + (x * ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333))))) / z_m;
	} else {
		tmp = x / ((z_m * y) / sin(y));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 0.0055d0) then
        tmp = (x + (x * ((y * y) * ((-0.16666666666666666d0) + ((y * y) * 0.008333333333333333d0))))) / z_m
    else
        tmp = x / ((z_m * y) / sin(y))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 0.0055) {
		tmp = (x + (x * ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333))))) / z_m;
	} else {
		tmp = x / ((z_m * y) / Math.sin(y));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 0.0055:
		tmp = (x + (x * ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333))))) / z_m
	else:
		tmp = x / ((z_m * y) / math.sin(y))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 0.0055)
		tmp = Float64(Float64(x + Float64(x * Float64(Float64(y * y) * Float64(-0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333))))) / z_m);
	else
		tmp = Float64(x / Float64(Float64(z_m * y) / sin(y)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 0.0055)
		tmp = (x + (x * ((y * y) * (-0.16666666666666666 + ((y * y) * 0.008333333333333333))))) / z_m;
	else
		tmp = x / ((z_m * y) / sin(y));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 0.0055], N[(N[(x + N[(x * N[(N[(y * y), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(x / N[(N[(z$95$m * y), $MachinePrecision] / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 0.0054999999999999997

   1. Initial program 95.5%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{\sin y}{y}\right), \color{blue}{z}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot \sin y}{y}\right), z\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot \sin y\right), y\right), z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \sin y\right), y\right), z\right) \]

      5. sin-lowering-sin.f64 84.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{sin.f64}\left(y\right)\right), y\right), z\right) \]

   3. Simplified 84.6%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot \sin y}{y}}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)}\right), y\right), z\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \left(1 + {y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)\right), y\right), z\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), y\right), z\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), y\right), z\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), y\right), z\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{120} \cdot {y}^{2} - \frac{1}{6}\right)\right)\right)\right)\right), y\right), z\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{120} \cdot {y}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right)\right), y\right), z\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{120} \cdot {y}^{2} + \frac{-1}{6}\right)\right)\right)\right)\right), y\right), z\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{-1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right), y\right), z\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\frac{1}{120} \cdot {y}^{2}\right)\right)\right)\right)\right)\right), y\right), z\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({y}^{2} \cdot \frac{1}{120}\right)\right)\right)\right)\right)\right), y\right), z\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), y\right), z\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), y\right), z\right) \]

      13. *-lowering-*.f64 57.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right)\right), y\right), z\right) \]

   7. Simplified 57.4%

      \[\leadsto \frac{\frac{x \cdot \color{blue}{\left(y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\right)}}{y}}{z} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{-1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right)}{y}\right), z\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{-1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right)}{y} \cdot x\right), z\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(1 + \left(y \cdot y\right) \cdot \left(\frac{-1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) \cdot y}{y} \cdot x\right), z\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(1 + \left(y \cdot y\right) \cdot \left(\frac{-1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) \cdot \frac{y}{y}\right) \cdot x\right), z\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(1 + \left(y \cdot y\right) \cdot \left(\frac{-1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) \cdot 1\right) \cdot x\right), z\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \left(y \cdot y\right) \cdot \left(\frac{-1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) \cdot \left(1 \cdot x\right)\right), z\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \left(y \cdot y\right) \cdot \left(\frac{-1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) \cdot x\right), z\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(\frac{-1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right)\right), z\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{-1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right) + 1\right)\right), z\right) \]

      10. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(y \cdot y\right) \cdot \left(\frac{-1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) \cdot x + 1 \cdot x\right), z\right) \]

      11. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(y \cdot y\right) \cdot \left(\frac{-1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) \cdot x + x\right), z\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(\left(y \cdot y\right) \cdot \left(\frac{-1}{6} + \left(y \cdot y\right) \cdot \frac{1}{120}\right)\right) \cdot x\right), x\right), z\right) \]

   9. Applied egg-rr 68.2%

      \[\leadsto \frac{\color{blue}{\left(\left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right) \cdot x + x}}{z} \]

   if 0.0054999999999999997 < y

   1. Initial program 92.2%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. associate-/l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(z \cdot \frac{y}{\sin y}\right)}\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{z \cdot y}{\color{blue}{\sin y}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(z \cdot y\right), \color{blue}{\sin y}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(y \cdot z\right), \sin \color{blue}{y}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \sin \color{blue}{y}\right)\right) \]

      9. sin-lowering-sin.f64 88.9%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{sin.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 88.9%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0055:\\ \;\;\;\;\frac{x + x \cdot \left(\left(y \cdot y\right) \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z \cdot y}{\sin y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z\_m \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{\sin y}{z\_m}}}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 1.6e-7)
    (/ x (* z_m (+ 1.0 (* (* y y) 0.16666666666666666))))
    (/ x (/ y (/ (sin y) z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1.6e-7) {
		tmp = x / (z_m * (1.0 + ((y * y) * 0.16666666666666666)));
	} else {
		tmp = x / (y / (sin(y) / z_m));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 1.6d-7) then
        tmp = x / (z_m * (1.0d0 + ((y * y) * 0.16666666666666666d0)))
    else
        tmp = x / (y / (sin(y) / z_m))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1.6e-7) {
		tmp = x / (z_m * (1.0 + ((y * y) * 0.16666666666666666)));
	} else {
		tmp = x / (y / (Math.sin(y) / z_m));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 1.6e-7:
		tmp = x / (z_m * (1.0 + ((y * y) * 0.16666666666666666)))
	else:
		tmp = x / (y / (math.sin(y) / z_m))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 1.6e-7)
		tmp = Float64(x / Float64(z_m * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666))));
	else
		tmp = Float64(x / Float64(y / Float64(sin(y) / z_m)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 1.6e-7)
		tmp = x / (z_m * (1.0 + ((y * y) * 0.16666666666666666)));
	else
		tmp = x / (y / (sin(y) / z_m));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 1.6e-7], N[(x / N[(z$95$m * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y / N[(N[Sin[y], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.6e-7

   1. Initial program 95.4%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. associate-/l/ N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(z \cdot \frac{y}{\sin y}\right)}\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{z \cdot y}{\color{blue}{\sin y}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(z \cdot y\right), \color{blue}{\sin y}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(y \cdot z\right), \sin \color{blue}{y}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \sin \color{blue}{y}\right)\right) \]

      9. sin-lowering-sin.f64 90.6%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{sin.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 90.6%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(z + \frac{1}{6} \cdot \left({y}^{2} \cdot z\right)\right)}\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(z + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{z}\right)\right) \]

      2. distribute-rgt1-in N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{z}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot z\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(z \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{6}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 79.3%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{6}\right)\right)\right)\right) \]

   7. Simplified 79.3%

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

   if 1.6e-7 < y

   1. Initial program 92.3%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sin y}{y}\right), \color{blue}{\left(\frac{z}{x}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\sin y, y\right), \left(\frac{\color{blue}{z}}{x}\right)\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right), \left(\frac{z}{x}\right)\right) \]

      7. /-lowering-/.f64 93.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), y\right), \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

   4. Applied egg-rr 93.0%

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

      1. associate-/r/ N/A

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

      2. clear-num N/A

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

      3. associate-/l/ N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(z \cdot \frac{y}{\sin y}\right)}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{y}{\sin y} \cdot \color{blue}{z}\right)\right) \]

      8. remove-double-div N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{y}{\sin y} \cdot \frac{1}{\color{blue}{\frac{1}{z}}}\right)\right) \]

      9. frac-times N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{y \cdot 1}{\color{blue}{\sin y \cdot \frac{1}{z}}}\right)\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{y}{\color{blue}{\sin y} \cdot \frac{1}{z}}\right)\right) \]

      11. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{y}{\frac{\sin y}{\color{blue}{z}}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{\sin y}{z}\right)}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\sin y, \color{blue}{z}\right)\right)\right) \]

      14. sin-lowering-sin.f64 89.1%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{sin.f64}\left(y\right), z\right)\right)\right) \]

   6. Applied egg-rr 89.1%

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

Alternative 5: 60.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \frac{1 + \left(y \cdot y\right) \cdot -0.16666666666666666}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z\_m \cdot y}{x}}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 8.6e+18)
    (* x (/ (+ 1.0 (* (* y y) -0.16666666666666666)) z_m))
    (/ y (/ (* z_m y) x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 8.6e+18) {
		tmp = x * ((1.0 + ((y * y) * -0.16666666666666666)) / z_m);
	} else {
		tmp = y / ((z_m * y) / x);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 8.6d+18) then
        tmp = x * ((1.0d0 + ((y * y) * (-0.16666666666666666d0))) / z_m)
    else
        tmp = y / ((z_m * y) / x)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 8.6e+18) {
		tmp = x * ((1.0 + ((y * y) * -0.16666666666666666)) / z_m);
	} else {
		tmp = y / ((z_m * y) / x);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 8.6e+18:
		tmp = x * ((1.0 + ((y * y) * -0.16666666666666666)) / z_m)
	else:
		tmp = y / ((z_m * y) / x)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 8.6e+18)
		tmp = Float64(x * Float64(Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)) / z_m));
	else
		tmp = Float64(y / Float64(Float64(z_m * y) / x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 8.6e+18)
		tmp = x * ((1.0 + ((y * y) * -0.16666666666666666)) / z_m);
	else
		tmp = y / ((z_m * y) / x);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 8.6e+18], N[(x * N[(N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(y / N[(N[(z$95$m * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 8.6e18

   1. Initial program 95.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(y \cdot \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right)}, y\right)\right), z\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \left(1 + \frac{-1}{6} \cdot {y}^{2}\right)\right), y\right)\right), z\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {y}^{2}\right)\right)\right), y\right)\right), z\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(y \cdot y\right)\right)\right)\right), y\right)\right), z\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(\left(\frac{-1}{6} \cdot y\right) \cdot y\right)\right)\right), y\right)\right), z\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \left(\frac{-1}{6} \cdot y\right)\right)\right)\right), y\right)\right), z\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(\frac{-1}{6} \cdot y\right)\right)\right)\right), y\right)\right), z\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \frac{-1}{6}\right)\right)\right)\right), y\right)\right), z\right) \]

      8. *-lowering-*.f64 65.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right)\right)\right), y\right)\right), z\right) \]

   5. Simplified 65.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot \left(1 + y \cdot \left(y \cdot \frac{-1}{6}\right)\right)}{y} \cdot x\right), z\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(1 + y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot y}{y} \cdot x\right), z\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(1 + y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \frac{y}{y}\right) \cdot x\right), z\right) \]

      4. *-inverses N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(1 + y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot 1\right) \cdot x\right), z\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot \left(1 \cdot x\right)\right), z\right) \]

      6. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot x\right), z\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(1 + y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right), z\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot \left(y \cdot \frac{-1}{6}\right) + 1\right)\right), z\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot x + 1 \cdot x\right), z\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot x + x\right), z\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right) \cdot x\right), x\right), z\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right), x\right), x\right), z\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \frac{-1}{6}\right)\right), x\right), x\right), z\right) \]

      14. *-lowering-*.f64 65.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{-1}{6}\right)\right), x\right), x\right), z\right) \]

   7. Applied egg-rr 65.9%

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

      1. /-rgt-identity N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. times-frac N/A

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

      7. /-rgt-identity N/A

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

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 + y \cdot \left(y \cdot \frac{-1}{6}\right)}{z}\right), \color{blue}{x}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(1 + y \cdot \left(y \cdot \frac{-1}{6}\right)\right), z\right), x\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(y \cdot \left(y \cdot \frac{-1}{6}\right)\right)\right), z\right), x\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{-1}{6}\right)\right), z\right), x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot \left(y \cdot y\right)\right)\right), z\right), x\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(y \cdot y\right)\right)\right), z\right), x\right) \]

      14. *-lowering-*.f64 65.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right), z\right), x\right) \]

   9. Applied egg-rr 65.6%

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

   if 8.6e18 < y

   1. Initial program 91.4%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 10.0%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 10.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{z}\right), \color{blue}{x}\right) \]

      4. /-lowering-/.f64 10.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, z\right), x\right) \]

   7. Applied egg-rr 10.0%

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

      1. associate-/r/ N/A

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

      2. clear-num N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{x}{z} \cdot {y}^{\color{blue}{0}} \]

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. inv-pow N/A

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

      8. associate-*l* N/A

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

      9. div-inv N/A

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

      10. *-commutative N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{y}{\frac{x}{z}}\right)}\right) \]

      14. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(\frac{y}{x} \cdot \color{blue}{z}\right)\right) \]

      15. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(\frac{y \cdot z}{\color{blue}{x}}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(y \cdot z\right), \color{blue}{x}\right)\right) \]

      17. *-lowering-*.f64 27.6%

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), x\right)\right) \]

   9. Applied egg-rr 27.6%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \frac{1 + \left(y \cdot y\right) \cdot -0.16666666666666666}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.0% accurate, 8.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z\_m \cdot y}{x}}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 3.5e-15) (/ x z_m) (/ y (/ (* z_m y) x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 3.5e-15) {
		tmp = x / z_m;
	} else {
		tmp = y / ((z_m * y) / x);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 3.5d-15) then
        tmp = x / z_m
    else
        tmp = y / ((z_m * y) / x)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 3.5e-15) {
		tmp = x / z_m;
	} else {
		tmp = y / ((z_m * y) / x);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 3.5e-15:
		tmp = x / z_m
	else:
		tmp = y / ((z_m * y) / x)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 3.5e-15)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(y / Float64(Float64(z_m * y) / x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 3.5e-15)
		tmp = x / z_m;
	else
		tmp = y / ((z_m * y) / x);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 3.5e-15], N[(x / z$95$m), $MachinePrecision], N[(y / N[(N[(z$95$m * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.5000000000000001e-15

   1. Initial program 95.4%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 71.7%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 71.7%

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

   if 3.5000000000000001e-15 < y

   1. Initial program 92.5%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 12.4%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 12.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{z}\right), \color{blue}{x}\right) \]

      4. /-lowering-/.f64 12.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, z\right), x\right) \]

   7. Applied egg-rr 12.4%

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

      1. associate-/r/ N/A

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

      2. clear-num N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{x}{z} \cdot {y}^{\color{blue}{0}} \]

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. inv-pow N/A

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

      8. associate-*l* N/A

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

      9. div-inv N/A

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

      10. *-commutative N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \color{blue}{\left(\frac{y}{\frac{x}{z}}\right)}\right) \]

      14. associate-/r/ N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(\frac{y}{x} \cdot \color{blue}{z}\right)\right) \]

      15. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(y, \left(\frac{y \cdot z}{\color{blue}{x}}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\left(y \cdot z\right), \color{blue}{x}\right)\right) \]

      17. *-lowering-*.f64 27.6%

          \[\leadsto \mathsf{/.f64}\left(y, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), x\right)\right) \]

   9. Applied egg-rr 27.6%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z\_m \cdot y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 2.9e-13) (/ x z_m) (* y (/ x (* z_m y))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 2.9e-13) {
		tmp = x / z_m;
	} else {
		tmp = y * (x / (z_m * y));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 2.9d-13) then
        tmp = x / z_m
    else
        tmp = y * (x / (z_m * y))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 2.9e-13) {
		tmp = x / z_m;
	} else {
		tmp = y * (x / (z_m * y));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 2.9e-13:
		tmp = x / z_m
	else:
		tmp = y * (x / (z_m * y))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 2.9e-13)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(y * Float64(x / Float64(z_m * y)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 2.9e-13)
		tmp = x / z_m;
	else
		tmp = y * (x / (z_m * y));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 2.9e-13], N[(x / z$95$m), $MachinePrecision], N[(y * N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.8999999999999998e-13

   1. Initial program 95.4%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 71.7%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 71.7%

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

   if 2.8999999999999998e-13 < y

   1. Initial program 92.5%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 12.4%

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

   5. Simplified 12.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{z}\right), \color{blue}{x}\right) \]

      4. /-lowering-/.f64 12.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, z\right), x\right) \]

   7. Applied egg-rr 12.4%

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

      1. associate-/r/ N/A

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

      2. clear-num N/A

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

      3. *-rgt-identity N/A

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

      4. metadata-eval N/A

         \[\leadsto \frac{x}{z} \cdot {y}^{\color{blue}{0}} \]

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. inv-pow N/A

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

      8. associate-*l* N/A

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

      9. div-inv N/A

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

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{x}{z}}{y}\right), \color{blue}{y}\right) \]

      11. associate-/l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{y \cdot z}\right), y\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y \cdot z\right)\right), y\right) \]

      13. *-lowering-*.f64 25.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), y\right) \]

   9. Applied egg-rr 25.5%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.6% accurate, 9.7× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (/ (/ x z_m) (+ 1.0 (* y (* y 0.16666666666666666))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	return z_s * ((x / z_m) / (1.0 + (y * (y * 0.16666666666666666))));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = z_s * ((x / z_m) / (1.0d0 + (y * (y * 0.16666666666666666d0))))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	return z_s * ((x / z_m) / (1.0 + (y * (y * 0.16666666666666666))));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	return z_s * ((x / z_m) / (1.0 + (y * (y * 0.16666666666666666))))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(Float64(x / z_m) / Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666)))))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m)
	tmp = z_s * ((x / z_m) / (1.0 + (y * (y * 0.16666666666666666))));
end

Wolfram

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

Derivation

1. Initial program 94.6%

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

   1. associate-*l/ N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{z}\right), \color{blue}{\left(\frac{y}{\sin y}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{\color{blue}{y}}{\sin y}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \color{blue}{\sin y}\right)\right) \]

   7. sin-lowering-sin.f64 97.3%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \mathsf{sin.f64}\left(y\right)\right)\right) \]

4. Applied egg-rr 97.3%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]
6. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right)\right) \]

   6. *-lowering-*.f64 64.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

7. Simplified 64.7%

   \[\leadsto \frac{\frac{x}{z}}{\color{blue}{1 + y \cdot \left(y \cdot 0.16666666666666666\right)}} \]
8. Add Preprocessing

Alternative 9: 66.4% accurate, 9.7× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (/ x (* z_m (+ 1.0 (* (* y y) 0.16666666666666666))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	return z_s * (x / (z_m * (1.0 + ((y * y) * 0.16666666666666666))));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = z_s * (x / (z_m * (1.0d0 + ((y * y) * 0.16666666666666666d0))))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	return z_s * (x / (z_m * (1.0 + ((y * y) * 0.16666666666666666))));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	return z_s * (x / (z_m * (1.0 + ((y * y) * 0.16666666666666666))))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(x / Float64(z_m * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666)))))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m)
	tmp = z_s * (x / (z_m * (1.0 + ((y * y) * 0.16666666666666666))));
end

Wolfram

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

Derivation

1. Initial program 94.6%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. associate-/l/ N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(z \cdot \frac{y}{\sin y}\right)}\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{z \cdot y}{\color{blue}{\sin y}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(z \cdot y\right), \color{blue}{\sin y}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(y \cdot z\right), \sin \color{blue}{y}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \sin \color{blue}{y}\right)\right) \]

   9. sin-lowering-sin.f64 90.1%

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{sin.f64}\left(y\right)\right)\right) \]

4. Applied egg-rr 90.1%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(z + \frac{1}{6} \cdot \left({y}^{2} \cdot z\right)\right)}\right) \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(z + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{z}\right)\right) \]

   2. distribute-rgt1-in N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\left(\frac{1}{6} \cdot {y}^{2} + 1\right) \cdot \color{blue}{z}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot z\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(z \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{6}}\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{6}\right)\right)\right)\right) \]

   10. *-lowering-*.f64 64.4%

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{6}\right)\right)\right)\right) \]

7. Simplified 64.4%

   \[\leadsto \frac{x}{\color{blue}{z \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)}} \]
8. Add Preprocessing

Alternative 10: 59.0% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.6%

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

3. Taylor expanded in y around 0

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

   1. /-lowering-/.f64 54.8%

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{z}\right) \]

5. Simplified 54.8%

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

Developer Target 1: 99.5% accurate, 0.9× 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 2024164 
(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))