Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 95.9% → 96.2%

Time: 10.0s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / (z / (sin(y) / y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. clear-num N/A

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

   6. div-inv N/A

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

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

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

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

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

   9. sin-lowering-sin.f64 98.4%

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

4. Applied egg-rr 98.4%

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

Alternative 2: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00019:\\ \;\;\;\;\frac{x}{z} \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.00019)
   (* (/ x z) (+ (* y (* y -0.16666666666666666)) 1.0))
   (* (sin y) (/ x (* z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00019) {
		tmp = (x / z) * ((y * (y * -0.16666666666666666)) + 1.0);
	} else {
		tmp = sin(y) * (x / (z * y));
	}
	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) :: tmp
    if (y <= 0.00019d0) then
        tmp = (x / z) * ((y * (y * (-0.16666666666666666d0))) + 1.0d0)
    else
        tmp = sin(y) * (x / (z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00019) {
		tmp = (x / z) * ((y * (y * -0.16666666666666666)) + 1.0);
	} else {
		tmp = Math.sin(y) * (x / (z * y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.00019:
		tmp = (x / z) * ((y * (y * -0.16666666666666666)) + 1.0)
	else:
		tmp = math.sin(y) * (x / (z * y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.00019)
		tmp = Float64(Float64(x / z) * Float64(Float64(y * Float64(y * -0.16666666666666666)) + 1.0));
	else
		tmp = Float64(sin(y) * Float64(x / Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.00019)
		tmp = (x / z) * ((y * (y * -0.16666666666666666)) + 1.0);
	else
		tmp = sin(y) * (x / (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.00019], N[(N[(x / z), $MachinePrecision] * N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(x / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9000000000000001e-4

   1. Initial program 96.7%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 86.5%

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

   3. Simplified 86.5%

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

   5. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 60.8%

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

   7. Simplified 60.8%

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

      1. associate-/l/ N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      11. /-lowering-/.f64 72.0%

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

   9. Applied egg-rr 72.0%

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

       1. *-inverses N/A

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

       2. *-commutative N/A

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

       3. *-lft-identity N/A

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

       4. +-commutative N/A

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

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

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

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

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

       7. *-lowering-*.f64 72.0%

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

   11. Applied egg-rr 72.0%

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

   if 1.9000000000000001e-4 < y

   1. Initial program 92.1%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 97.0%

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

   4. Applied egg-rr 97.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00019:\\ \;\;\;\;\frac{x}{z} \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{x}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.15e+30)
   (* (/ x z) (+ (* y (* y -0.16666666666666666)) 1.0))
   (* y (/ x (* z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.15e+30) {
		tmp = (x / z) * ((y * (y * -0.16666666666666666)) + 1.0);
	} else {
		tmp = y * (x / (z * y));
	}
	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) :: tmp
    if (y <= 1.15d+30) then
        tmp = (x / z) * ((y * (y * (-0.16666666666666666d0))) + 1.0d0)
    else
        tmp = y * (x / (z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.15e+30) {
		tmp = (x / z) * ((y * (y * -0.16666666666666666)) + 1.0);
	} else {
		tmp = y * (x / (z * y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.15e+30:
		tmp = (x / z) * ((y * (y * -0.16666666666666666)) + 1.0)
	else:
		tmp = y * (x / (z * y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.15e+30)
		tmp = Float64(Float64(x / z) * Float64(Float64(y * Float64(y * -0.16666666666666666)) + 1.0));
	else
		tmp = Float64(y * Float64(x / Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.15e+30)
		tmp = (x / z) * ((y * (y * -0.16666666666666666)) + 1.0);
	else
		tmp = y * (x / (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.15e+30], N[(N[(x / z), $MachinePrecision] * N[(N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.15e30

   1. Initial program 96.9%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 87.4%

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

   3. Simplified 87.4%

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

   5. Taylor expanded in y around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 57.8%

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

   7. Simplified 57.8%

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

      1. associate-/l/ N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      11. /-lowering-/.f64 68.2%

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

   9. Applied egg-rr 68.2%

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

       1. *-inverses N/A

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

       2. *-commutative N/A

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

       3. *-lft-identity N/A

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

       4. +-commutative N/A

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

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

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

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

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

       7. *-lowering-*.f64 68.2%

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

   11. Applied egg-rr 68.2%

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

   if 1.15e30 < y

   1. Initial program 90.2%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 96.5%

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

   4. Applied egg-rr 96.5%

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

   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 26.8%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y \cdot \left(y \cdot -0.16666666666666666\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right) \cdot \frac{z}{x}} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ 1.0 (* (+ 1.0 (* (* y y) 0.16666666666666666)) (/ z x))))

C

double code(double x, double y, double z) {
	return 1.0 / ((1.0 + ((y * y) * 0.16666666666666666)) * (z / x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 / ((1.0d0 + ((y * y) * 0.16666666666666666d0)) * (z / x))
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 / ((1.0 + ((y * y) * 0.16666666666666666)) * (z / x));
}

Python

def code(x, y, z):
	return 1.0 / ((1.0 + ((y * y) * 0.16666666666666666)) * (z / x))

Julia

function code(x, y, z)
	return Float64(1.0 / Float64(Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666)) * Float64(z / x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 / ((1.0 + ((y * y) * 0.16666666666666666)) * (z / x));
end

Wolfram

code[x_, y_, z_] := N[(1.0 / N[(N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.4%

   \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
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 88.0%

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

3. Simplified 88.0%

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

5. Taylor expanded in y around 0

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

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

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

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

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 45.2%

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

7. Simplified 45.2%

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

   1. clear-num N/A

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

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

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

   3. clear-num N/A

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

   4. associate-/l/ N/A

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

   5. clear-num N/A

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

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

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

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

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

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

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

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

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

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

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

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

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

   14. *-lowering-*.f64 39.7%

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

9. Applied egg-rr 39.7%

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

10. Taylor expanded in y around 0

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

    1. associate-/l* N/A

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

    2. associate-*r* N/A

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

    3. distribute-lft1-in N/A

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

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

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

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

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

    6. *-commutative N/A

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

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

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

    8. unpow2 N/A

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

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

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

    10. /-lowering-/.f64 67.6%

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

12. Simplified 67.6%

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

13. Final simplification 67.6%

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

Alternative 5: 62.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00095:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.00095) (/ x z) (* y (/ x (* z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00095) {
		tmp = x / z;
	} else {
		tmp = y * (x / (z * y));
	}
	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) :: tmp
    if (y <= 0.00095d0) then
        tmp = x / z
    else
        tmp = y * (x / (z * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.00095) {
		tmp = x / z;
	} else {
		tmp = y * (x / (z * y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.00095:
		tmp = x / z
	else:
		tmp = y * (x / (z * y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.00095)
		tmp = Float64(x / z);
	else
		tmp = Float64(y * Float64(x / Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.00095)
		tmp = x / z;
	else
		tmp = y * (x / (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.00095], N[(x / z), $MachinePrecision], N[(y * N[(x / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.49999999999999998e-4

   1. Initial program 96.7%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 76.7%

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

   5. Simplified 76.7%

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

   if 9.49999999999999998e-4 < y

   1. Initial program 92.1%

      \[\frac{x \cdot \frac{\sin y}{y}}{z} \]
   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 97.0%

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

   4. Applied egg-rr 97.0%

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

   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 24.0%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00095:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{z \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ x (* z (+ 1.0 (* (* y y) 0.16666666666666666)))))

C

double code(double x, double y, double z) {
	return x / (z * (1.0 + ((y * y) * 0.16666666666666666)));
}

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 / (z * (1.0d0 + ((y * y) * 0.16666666666666666d0)))
end function

Java

public static double code(double x, double y, double z) {
	return x / (z * (1.0 + ((y * y) * 0.16666666666666666)));
}

Python

def code(x, y, z):
	return x / (z * (1.0 + ((y * y) * 0.16666666666666666)))

Julia

function code(x, y, z)
	return Float64(x / Float64(z * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x / (z * (1.0 + ((y * y) * 0.16666666666666666)));
end

Wolfram

code[x_, y_, z_] := N[(x / N[(z * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. clear-num N/A

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

   6. div-inv N/A

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

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

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

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

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

   9. sin-lowering-sin.f64 98.4%

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

4. Applied egg-rr 98.4%

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

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. *-lowering-*.f64 N/A

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

   4. +-commutative N/A

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

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

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 67.4%

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

7. Simplified 67.4%

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

8. Final simplification 67.4%

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

Alternative 7: 58.5% accurate, 35.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x / 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 / z
end function

Java

public static double code(double x, double y, double z) {
	return x / z;
}

Python

def code(x, y, z):
	return x / z

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 59.6%

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

5. Simplified 59.6%

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

Developer Target 1: 99.6% 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 2024158 
(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))