Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.3% → 99.6%

Time: 10.3s

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: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.3 \cdot 10^{+37}:\\ \;\;\;\;\frac{x}{z\_m \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z\_m}\\ \end{array} \end{array} \]

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 3.3e+37)
    (/ x (* z_m (/ y (sin y))))
    (/ (* 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 <= 3.3e+37) {
		tmp = x / (z_m * (y / sin(y)));
	} 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 <= 3.3d+37) then
        tmp = x / (z_m * (y / sin(y)))
    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 <= 3.3e+37) {
		tmp = x / (z_m * (y / Math.sin(y)));
	} 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 <= 3.3e+37:
		tmp = x / (z_m * (y / math.sin(y)))
	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 <= 3.3e+37)
		tmp = Float64(x / Float64(z_m * Float64(y / sin(y))));
	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 <= 3.3e+37)
		tmp = x / (z_m * (y / sin(y)));
	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, 3.3e+37], N[(x / N[(z$95$m * N[(y / N[Sin[y], $MachinePrecision]), $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 < 3.3000000000000001e37

   1. Initial program 92.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 91.5%

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

   4. Applied egg-rr 91.5%

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

      1. associate-*l/ N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

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

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

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

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

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

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

      9. sin-lowering-sin.f64 98.1%

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

   6. Applied egg-rr 98.1%

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

   if 3.3000000000000001e37 < z

   1. Initial program 99.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.3 \cdot 10^{+37}:\\ \;\;\;\;\frac{x}{z \cdot \frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\sin y}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{z\_m \cdot y}\\ \end{array} \end{array} \]

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 2.7e-14) (/ x z_m) (* (sin 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 <= 2.7e-14) {
		tmp = x / z_m;
	} else {
		tmp = sin(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 <= 2.7d-14) then
        tmp = x / z_m
    else
        tmp = sin(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 <= 2.7e-14) {
		tmp = x / z_m;
	} else {
		tmp = Math.sin(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 <= 2.7e-14:
		tmp = x / z_m
	else:
		tmp = math.sin(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 <= 2.7e-14)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(sin(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 <= 2.7e-14)
		tmp = x / z_m;
	else
		tmp = sin(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, 2.7e-14], N[(x / z$95$m), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.6999999999999999e-14

   1. Initial program 95.9%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 69.3%

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

   5. Simplified 69.3%

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

   if 2.6999999999999999e-14 < y

   1. Initial program 88.6%

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. sin-lowering-sin.f64 94.5%

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

   4. Applied egg-rr 94.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 1.0× speedup

Math

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

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 (/ 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) {
	return z_s * (x / (z_m * (y / sin(y))));
}

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

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

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(y / sin(y)))))
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 * (y / sin(y))));
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[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.8%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. associate-/l/ N/A

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 90.8%

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

4. Applied egg-rr 90.8%

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

   1. associate-*l/ N/A

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

   2. times-frac N/A

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

   3. clear-num N/A

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

   4. frac-times N/A

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

   5. *-lft-identity N/A

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

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

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

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

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

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

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

   9. sin-lowering-sin.f64 96.1%

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

6. Applied egg-rr 96.1%

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

7. Final simplification 96.1%

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

Alternative 4: 61.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x \cdot 6}{z\_m}}{y}}{y}\\ \end{array} \end{array} \]

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 8e+32)
    (* (/ x z_m) (+ 1.0 (* y (* y -0.16666666666666666))))
    (/ (/ (/ (* x 6.0) z_m) y) 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 <= 8e+32) {
		tmp = (x / z_m) * (1.0 + (y * (y * -0.16666666666666666)));
	} else {
		tmp = (((x * 6.0) / z_m) / y) / 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 <= 8d+32) then
        tmp = (x / z_m) * (1.0d0 + (y * (y * (-0.16666666666666666d0))))
    else
        tmp = (((x * 6.0d0) / z_m) / y) / 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 <= 8e+32) {
		tmp = (x / z_m) * (1.0 + (y * (y * -0.16666666666666666)));
	} else {
		tmp = (((x * 6.0) / z_m) / y) / 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 <= 8e+32:
		tmp = (x / z_m) * (1.0 + (y * (y * -0.16666666666666666)))
	else:
		tmp = (((x * 6.0) / z_m) / y) / 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 <= 8e+32)
		tmp = Float64(Float64(x / z_m) * Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666))));
	else
		tmp = Float64(Float64(Float64(Float64(x * 6.0) / z_m) / y) / 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 <= 8e+32)
		tmp = (x / z_m) * (1.0 + (y * (y * -0.16666666666666666)));
	else
		tmp = (((x * 6.0) / z_m) / y) / 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, 8e+32], N[(N[(x / z$95$m), $MachinePrecision] * N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * 6.0), $MachinePrecision] / z$95$m), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 8.00000000000000043e32

   1. Initial program 96.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. distribute-lft1-in N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. *-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

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

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

   5. Simplified 64.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. /-lowering-/.f64 66.1%

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

   7. Applied egg-rr 66.1%

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

   if 8.00000000000000043e32 < y

   1. Initial program 87.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 94.0%

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

   4. Applied egg-rr 94.0%

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

      1. associate-*l/ N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

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

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

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

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

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

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

      9. sin-lowering-sin.f64 94.0%

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

   6. Applied egg-rr 94.0%

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

   7. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 35.3%

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

   9. Simplified 35.3%

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

   10. Taylor expanded in y around inf

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

       1. associate-*r/ N/A

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

       2. associate-/l/ N/A

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

       3. associate-*r/ N/A

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

       4. unpow2 N/A

          \[\leadsto \frac{6 \cdot \frac{x}{z}}{y \cdot \color{blue}{y}} \]

       5. associate-/r* N/A

          \[\leadsto \frac{\frac{6 \cdot \frac{x}{z}}{y}}{\color{blue}{y}} \]

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

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

       7. /-lowering-/.f64 N/A

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

       8. associate-*r/ N/A

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

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

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 36.5%

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

   12. Simplified 36.5%

       \[\leadsto \color{blue}{\frac{\frac{\frac{x \cdot 6}{z}}{y}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+32}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + y \cdot \left(y \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x \cdot 6}{z}}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.8% accurate, 7.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x \cdot 6}{z\_m}}{y}}{y}\\ \end{array} \end{array} \]

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 2.7e-12) (/ x z_m) (/ (/ (/ (* x 6.0) z_m) y) 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 <= 2.7e-12) {
		tmp = x / z_m;
	} else {
		tmp = (((x * 6.0) / z_m) / y) / 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 <= 2.7d-12) then
        tmp = x / z_m
    else
        tmp = (((x * 6.0d0) / z_m) / y) / 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 <= 2.7e-12) {
		tmp = x / z_m;
	} else {
		tmp = (((x * 6.0) / z_m) / y) / 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 <= 2.7e-12:
		tmp = x / z_m
	else:
		tmp = (((x * 6.0) / z_m) / y) / 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 <= 2.7e-12)
		tmp = Float64(x / z_m);
	else
		tmp = Float64(Float64(Float64(Float64(x * 6.0) / z_m) / y) / 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 <= 2.7e-12)
		tmp = x / z_m;
	else
		tmp = (((x * 6.0) / z_m) / y) / 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, 2.7e-12], N[(x / z$95$m), $MachinePrecision], N[(N[(N[(N[(x * 6.0), $MachinePrecision] / z$95$m), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.6999999999999998e-12

   1. Initial program 95.9%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 69.4%

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

   5. Simplified 69.4%

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

   if 2.6999999999999998e-12 < y

   1. Initial program 88.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-/l/ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 94.5%

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

   4. Applied egg-rr 94.5%

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

      1. associate-*l/ N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

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

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

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

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

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

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

      9. sin-lowering-sin.f64 94.6%

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

   6. Applied egg-rr 94.6%

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

   7. Taylor expanded in y around 0

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

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

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 34.9%

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

   9. Simplified 34.9%

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

   10. Taylor expanded in y around inf

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

       1. associate-*r/ N/A

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

       2. associate-/l/ N/A

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

       3. associate-*r/ N/A

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

       4. unpow2 N/A

          \[\leadsto \frac{6 \cdot \frac{x}{z}}{y \cdot \color{blue}{y}} \]

       5. associate-/r* N/A

          \[\leadsto \frac{\frac{6 \cdot \frac{x}{z}}{y}}{\color{blue}{y}} \]

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

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

       7. /-lowering-/.f64 N/A

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

       8. associate-*r/ N/A

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

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

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

       10. *-commutative N/A

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

       11. *-lowering-*.f64 36.0%

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

   12. Simplified 36.0%

       \[\leadsto \color{blue}{\frac{\frac{\frac{x \cdot 6}{z}}{y}}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 66.4% accurate, 8.2× speedup

Math

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

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 (/ -1.0 (* (+ -1.0 (* y (* y 0.16666666666666666))) (/ z_m x)))))

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 * (-1.0 / ((-1.0 + (y * (y * 0.16666666666666666))) * (z_m / x)));
}

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 * ((-1.0d0) / (((-1.0d0) + (y * (y * 0.16666666666666666d0))) * (z_m / x)))
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 * (-1.0 / ((-1.0 + (y * (y * 0.16666666666666666))) * (z_m / x)));
}

Python

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

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(-1.0 / Float64(Float64(-1.0 + Float64(y * Float64(y * 0.16666666666666666))) * Float64(z_m / x))))
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 * (-1.0 / ((-1.0 + (y * (y * 0.16666666666666666))) * (z_m / x)));
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[(-1.0 / N[(N[(-1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.8%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. associate-/l/ N/A

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 90.8%

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

4. Applied egg-rr 90.8%

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

   1. associate-*l/ N/A

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

   2. times-frac N/A

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

   3. clear-num N/A

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

   4. frac-times N/A

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

   5. *-lft-identity N/A

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

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

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

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

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

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

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

   9. sin-lowering-sin.f64 96.1%

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

6. Applied egg-rr 96.1%

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

7. Taylor expanded in y around 0

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

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

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

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

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

   6. *-lowering-*.f64 64.1%

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

9. Simplified 64.1%

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

11. Applied egg-rr 65.1%

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

12. Final simplification 65.1%

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

Alternative 7: 66.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(0 - \frac{\frac{x}{-1 + y \cdot \left(y \cdot 0.16666666666666666\right)}}{z\_m}\right) \end{array} \]

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 (- 0.0 (/ (/ x (+ -1.0 (* y (* y 0.16666666666666666)))) 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 * (0.0 - ((x / (-1.0 + (y * (y * 0.16666666666666666)))) / 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 * (0.0d0 - ((x / ((-1.0d0) + (y * (y * 0.16666666666666666d0)))) / 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 * (0.0 - ((x / (-1.0 + (y * (y * 0.16666666666666666)))) / 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 * (0.0 - ((x / (-1.0 + (y * (y * 0.16666666666666666)))) / 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(0.0 - Float64(Float64(x / Float64(-1.0 + Float64(y * Float64(y * 0.16666666666666666)))) / 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 * (0.0 - ((x / (-1.0 + (y * (y * 0.16666666666666666)))) / 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[(0.0 - N[(N[(x / N[(-1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.8%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. associate-/l/ N/A

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 90.8%

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

4. Applied egg-rr 90.8%

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

   1. associate-*l/ N/A

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

   2. times-frac N/A

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

   3. clear-num N/A

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

   4. frac-times N/A

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

   5. *-lft-identity N/A

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

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

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

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

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

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

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

   9. sin-lowering-sin.f64 96.1%

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

6. Applied egg-rr 96.1%

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

7. Taylor expanded in y around 0

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

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

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

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

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

   6. *-lowering-*.f64 64.1%

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

9. Simplified 64.1%

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

11. Applied egg-rr 65.1%

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

12. Final simplification 65.1%

    \[\leadsto 0 - \frac{\frac{x}{-1 + y \cdot \left(y \cdot 0.16666666666666666\right)}}{z} \]
13. Add Preprocessing

Alternative 8: 62.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z\_m \cdot y}\\ \end{array} \end{array} \]

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 2.7e-14) (/ 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 <= 2.7e-14) {
		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 <= 2.7d-14) 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 <= 2.7e-14) {
		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 <= 2.7e-14:
		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 <= 2.7e-14)
		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 <= 2.7e-14)
		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, 2.7e-14], 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 < 2.6999999999999999e-14

   1. Initial program 95.9%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 69.3%

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

   5. Simplified 69.3%

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

   if 2.6999999999999999e-14 < y

   1. Initial program 88.6%

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

      1. associate-*r/ N/A

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

      2. associate-/l/ N/A

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

      3. associate-*l/ N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. sin-lowering-sin.f64 94.5%

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

   4. Applied egg-rr 94.5%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \color{blue}{y}\right) \]
   6. Step-by-step derivation

   7. Simplified 35.8%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.4% accurate, 9.7× speedup

Math

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

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(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[(x / N[(z$95$m * N[(1.0 - N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.8%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. associate-/l/ N/A

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 90.8%

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

4. Applied egg-rr 90.8%

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

   1. associate-*l/ N/A

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

   2. times-frac N/A

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

   3. clear-num N/A

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

   4. frac-times N/A

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

   5. *-lft-identity N/A

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

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

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

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

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

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

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

   9. sin-lowering-sin.f64 96.1%

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

6. Applied egg-rr 96.1%

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

7. Taylor expanded in y around 0

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

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

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

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

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

   6. *-lowering-*.f64 64.1%

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

9. Simplified 64.1%

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

11. Applied egg-rr 65.0%

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

12. Final simplification 65.0%

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

Alternative 10: 58.8% 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 93.8%

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

3. Taylor expanded in y around 0

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

   1. /-lowering-/.f64 55.9%

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

5. Simplified 55.9%

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

Developer Target 1: 99.3% 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 2024138 
(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))