Linear.Quaternion:$ctanh from linear-1.19.1.3

Percentage Accurate: 96.0% → 95.9%

Time: 8.1s

Alternatives: 9

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

   1. associate-*l/ 97.2%

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

   2. *-commutative 97.2%

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

3. Applied egg-rr 97.2%

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

4. Final simplification 97.2%

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

Alternative 2: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-8} \lor \neg \left(y \leq 10^{-31}\right):\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.7e-8) (not (<= y 1e-31))) (* (sin y) (/ x (* y z))) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.7e-8) || !(y <= 1e-31)) {
		tmp = sin(y) * (x / (y * z));
	} else {
		tmp = x / z;
	}
	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 <= (-2.7d-8)) .or. (.not. (y <= 1d-31))) then
        tmp = sin(y) * (x / (y * z))
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.7e-8) || !(y <= 1e-31)) {
		tmp = Math.sin(y) * (x / (y * z));
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.7e-8) or not (y <= 1e-31):
		tmp = math.sin(y) * (x / (y * z))
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.7e-8) || !(y <= 1e-31))
		tmp = Float64(sin(y) * Float64(x / Float64(y * z)));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.7e-8) || ~((y <= 1e-31)))
		tmp = sin(y) * (x / (y * z));
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.7e-8], N[Not[LessEqual[y, 1e-31]], $MachinePrecision]], N[(N[Sin[y], $MachinePrecision] * N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.70000000000000002e-8 or 1e-31 < y

   1. Initial program 92.0%

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

      1. associate-*l/ 94.8%

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

      2. times-frac 89.3%

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

      3. *-commutative 89.3%

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

      4. associate-*r/ 89.2%

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

      5. *-commutative 89.2%

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

   3. Simplified 89.2%

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

   if -2.70000000000000002e-8 < y < 1e-31

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-8} \lor \neg \left(y \leq 10^{-31}\right):\\ \;\;\;\;\sin y \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 3: 65.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+67}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;y \leq 2.3:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.4e+67)
   (* 6.0 (/ x (* z (* y y))))
   (if (<= y 2.3)
     (* (/ x z) (+ 1.0 (* (* y y) -0.16666666666666666)))
     (/ y (* z (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.4e+67) {
		tmp = 6.0 * (x / (z * (y * y)));
	} else if (y <= 2.3) {
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = y / (z * (y / x));
	}
	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 <= (-4.4d+67)) then
        tmp = 6.0d0 * (x / (z * (y * y)))
    else if (y <= 2.3d0) then
        tmp = (x / z) * (1.0d0 + ((y * y) * (-0.16666666666666666d0)))
    else
        tmp = y / (z * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.4e+67) {
		tmp = 6.0 * (x / (z * (y * y)));
	} else if (y <= 2.3) {
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666));
	} else {
		tmp = y / (z * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.4e+67:
		tmp = 6.0 * (x / (z * (y * y)))
	elif y <= 2.3:
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666))
	else:
		tmp = y / (z * (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.4e+67)
		tmp = Float64(6.0 * Float64(x / Float64(z * Float64(y * y))));
	elseif (y <= 2.3)
		tmp = Float64(Float64(x / z) * Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666)));
	else
		tmp = Float64(y / Float64(z * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.4e+67)
		tmp = 6.0 * (x / (z * (y * y)));
	elseif (y <= 2.3)
		tmp = (x / z) * (1.0 + ((y * y) * -0.16666666666666666));
	else
		tmp = y / (z * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.4e+67], N[(6.0 * N[(x / N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3], N[(N[(x / z), $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.4e67

   1. Initial program 91.7%

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

   2. Taylor expanded in y around 0 2.1%

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {y}^{2}\right)}}{z} \]
   3. Step-by-step derivation

      1. unpow2 2.1%

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

   4. Simplified 2.1%

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

      1. distribute-rgt-in 2.1%

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

      2. *-un-lft-identity 2.1%

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

      3. flip-+ 0.5%

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

      4. *-commutative 0.5%

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

      5. *-commutative 0.5%

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

      6. associate-*r* 0.5%

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

      7. associate-*r* 0.5%

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

      8. *-commutative 0.5%

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

      9. associate-*r* 0.5%

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

      10. associate-*r* 4.4%

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

      11. *-commutative 4.4%

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

      12. *-commutative 4.4%

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

      13. associate-*r* 4.4%

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

      14. associate-*r* 1.2%

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

      15. *-commutative 1.2%

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

   6. Applied egg-rr 1.2%

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

   7. Taylor expanded in y around 0 33.9%

      \[\leadsto \frac{\frac{\color{blue}{{x}^{2}}}{x - \left(x \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot y}}{z} \]
   8. Step-by-step derivation

      1. unpow2 33.9%

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

   9. Simplified 33.9%

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

   10. Taylor expanded in y around inf 36.8%

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

       1. unpow2 36.8%

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

       2. *-commutative 36.8%

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

   12. Simplified 36.8%

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

   if -4.4e67 < y < 2.2999999999999998

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in y around 0 95.0%

      \[\leadsto \color{blue}{\left(1 + -0.16666666666666666 \cdot {y}^{2}\right)} \cdot \frac{x}{z} \]
   5. Step-by-step derivation

      1. unpow2 94.9%

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

   6. Simplified 95.0%

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

   if 2.2999999999999998 < y

   1. Initial program 89.6%

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

      1. associate-*r/ 89.7%

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

      2. associate-/l/ 88.6%

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

      3. *-commutative 88.6%

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

      4. times-frac 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in y around 0 30.9%

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

      1. *-commutative 30.9%

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

      2. clear-num 30.9%

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

      3. frac-times 36.3%

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

      4. *-un-lft-identity 36.3%

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

   6. Applied egg-rr 36.3%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+67}:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;y \leq 2.3:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 + \left(y \cdot y\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 4: 65.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+22} \lor \neg \left(y \leq 3.5 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6e+22) (not (<= y 3.5e-23))) (/ y (* z (/ y x))) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6e+22) || !(y <= 3.5e-23)) {
		tmp = y / (z * (y / x));
	} else {
		tmp = x / z;
	}
	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 <= (-6d+22)) .or. (.not. (y <= 3.5d-23))) then
        tmp = y / (z * (y / x))
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6e+22) || !(y <= 3.5e-23)) {
		tmp = y / (z * (y / x));
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -6e+22) or not (y <= 3.5e-23):
		tmp = y / (z * (y / x))
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6e+22) || !(y <= 3.5e-23))
		tmp = Float64(y / Float64(z * Float64(y / x)));
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -6e+22) || ~((y <= 3.5e-23)))
		tmp = y / (z * (y / x));
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6e+22], N[Not[LessEqual[y, 3.5e-23]], $MachinePrecision]], N[(y / N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6e22 or 3.49999999999999993e-23 < y

   1. Initial program 91.6%

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

      1. associate-*r/ 91.7%

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

      2. associate-/l/ 88.8%

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

      3. *-commutative 88.8%

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

      4. times-frac 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in y around 0 29.5%

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

      1. *-commutative 29.5%

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

      2. clear-num 30.2%

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

      3. frac-times 38.3%

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

      4. *-un-lft-identity 38.3%

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

   6. Applied egg-rr 38.3%

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

   if -6e22 < y < 3.49999999999999993e-23

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+22} \lor \neg \left(y \leq 3.5 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 5: 61.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{\frac{y \cdot z}{y}}\\ \mathbf{elif}\;y \leq 400000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.2e+90)
   (/ x (/ (* y z) y))
   (if (<= y 400000.0) (/ x z) (* (/ y z) (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.2e+90) {
		tmp = x / ((y * z) / y);
	} else if (y <= 400000.0) {
		tmp = x / z;
	} else {
		tmp = (y / z) * (x / 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 <= (-4.2d+90)) then
        tmp = x / ((y * z) / y)
    else if (y <= 400000.0d0) then
        tmp = x / z
    else
        tmp = (y / z) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.2e+90) {
		tmp = x / ((y * z) / y);
	} else if (y <= 400000.0) {
		tmp = x / z;
	} else {
		tmp = (y / z) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.2e+90:
		tmp = x / ((y * z) / y)
	elif y <= 400000.0:
		tmp = x / z
	else:
		tmp = (y / z) * (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.2e+90)
		tmp = Float64(x / Float64(Float64(y * z) / y));
	elseif (y <= 400000.0)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(y / z) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4.2e+90)
		tmp = x / ((y * z) / y);
	elseif (y <= 400000.0)
		tmp = x / z;
	else
		tmp = (y / z) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.2e+90], N[(x / N[(N[(y * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 400000.0], N[(x / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.19999999999999961e90

   1. Initial program 91.1%

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

      1. associate-/l* 86.7%

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

      2. associate-/r/ 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in y around 0 19.5%

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

      1. associate-*l/ 29.3%

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

      2. *-commutative 29.3%

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

   6. Applied egg-rr 29.3%

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

   if -4.19999999999999961e90 < y < 4e5

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 89.4%

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

   if 4e5 < y

   1. Initial program 89.1%

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

      1. associate-*r/ 89.2%

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

      2. associate-/l/ 88.1%

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

      3. *-commutative 88.1%

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

      4. times-frac 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in y around 0 30.6%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+90}:\\ \;\;\;\;\frac{x}{\frac{y \cdot z}{y}}\\ \mathbf{elif}\;y \leq 400000:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 6: 65.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.4)
   (* 6.0 (/ x (* z (* y y))))
   (if (<= y 3.5e-23) (/ x z) (/ y (* z (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4) {
		tmp = 6.0 * (x / (z * (y * y)));
	} else if (y <= 3.5e-23) {
		tmp = x / z;
	} else {
		tmp = y / (z * (y / x));
	}
	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 <= (-2.4d0)) then
        tmp = 6.0d0 * (x / (z * (y * y)))
    else if (y <= 3.5d-23) then
        tmp = x / z
    else
        tmp = y / (z * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4) {
		tmp = 6.0 * (x / (z * (y * y)));
	} else if (y <= 3.5e-23) {
		tmp = x / z;
	} else {
		tmp = y / (z * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.4:
		tmp = 6.0 * (x / (z * (y * y)))
	elif y <= 3.5e-23:
		tmp = x / z
	else:
		tmp = y / (z * (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.4)
		tmp = Float64(6.0 * Float64(x / Float64(z * Float64(y * y))));
	elseif (y <= 3.5e-23)
		tmp = Float64(x / z);
	else
		tmp = Float64(y / Float64(z * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.4)
		tmp = 6.0 * (x / (z * (y * y)));
	elseif (y <= 3.5e-23)
		tmp = x / z;
	else
		tmp = y / (z * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.4], N[(6.0 * N[(x / N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e-23], N[(x / z), $MachinePrecision], N[(y / N[(z * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.39999999999999991

   1. Initial program 92.7%

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

   2. Taylor expanded in y around 0 3.8%

      \[\leadsto \frac{x \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {y}^{2}\right)}}{z} \]
   3. Step-by-step derivation

      1. unpow2 3.8%

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

   4. Simplified 3.8%

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

      1. distribute-rgt-in 3.8%

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

      2. *-un-lft-identity 3.8%

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

      3. flip-+ 0.8%

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

      4. *-commutative 0.8%

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

      5. *-commutative 0.8%

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

      6. associate-*r* 0.8%

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

      7. associate-*r* 0.8%

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

      8. *-commutative 0.8%

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

      9. associate-*r* 0.8%

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

      10. associate-*r* 4.1%

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

      11. *-commutative 4.1%

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

      12. *-commutative 4.1%

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

      13. associate-*r* 4.2%

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

      14. associate-*r* 1.3%

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

      15. *-commutative 1.3%

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

   6. Applied egg-rr 1.3%

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

   7. Taylor expanded in y around 0 30.7%

      \[\leadsto \frac{\frac{\color{blue}{{x}^{2}}}{x - \left(x \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot y}}{z} \]
   8. Step-by-step derivation

      1. unpow2 30.7%

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

   9. Simplified 30.7%

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

   10. Taylor expanded in y around inf 33.4%

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

       1. unpow2 33.4%

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

       2. *-commutative 33.4%

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

   12. Simplified 33.4%

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

   if -2.39999999999999991 < y < 3.49999999999999993e-23

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 99.4%

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

   if 3.49999999999999993e-23 < y

   1. Initial program 90.8%

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

      1. associate-*r/ 90.9%

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

      2. associate-/l/ 90.0%

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

      3. *-commutative 90.0%

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

      4. times-frac 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in y around 0 37.8%

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

      1. *-commutative 37.8%

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

      2. clear-num 37.8%

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

      3. frac-times 42.6%

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

      4. *-un-lft-identity 42.6%

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

   6. Applied egg-rr 42.6%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4:\\ \;\;\;\;6 \cdot \frac{x}{z \cdot \left(y \cdot y\right)}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 7: 66.1% accurate, 9.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x / (z + (z * (0.16666666666666666 * (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 + (z * (0.16666666666666666d0 * (y * y))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.7%

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

2. Taylor expanded in y around 0 53.1%

   \[\leadsto \frac{x \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {y}^{2}\right)}}{z} \]
3. Step-by-step derivation

   1. unpow2 53.1%

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

4. Simplified 53.1%

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

   1. distribute-rgt-in 53.1%

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

   2. *-un-lft-identity 53.1%

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

   3. flip-+ 34.7%

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

   4. *-commutative 34.7%

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

   5. *-commutative 34.7%

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

   6. associate-*r* 34.7%

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

   7. associate-*r* 34.7%

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

   8. *-commutative 34.7%

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

   9. associate-*r* 34.7%

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

   10. associate-*r* 36.0%

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

   11. *-commutative 36.0%

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

   12. *-commutative 36.0%

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

   13. associate-*r* 36.0%

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

   14. associate-*r* 34.9%

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

   15. *-commutative 34.9%

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

6. Applied egg-rr 34.9%

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

7. Taylor expanded in y around 0 49.9%

   \[\leadsto \frac{\frac{\color{blue}{{x}^{2}}}{x - \left(x \cdot \left(y \cdot -0.16666666666666666\right)\right) \cdot y}}{z} \]
8. Step-by-step derivation

   1. unpow2 49.9%

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

9. Simplified 49.9%

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

10. Taylor expanded in x around 0 68.0%

    \[\leadsto \color{blue}{\frac{x}{\left(1 - -0.16666666666666666 \cdot {y}^{2}\right) \cdot z}} \]
11. Step-by-step derivation

    1. *-commutative 68.0%

       \[\leadsto \frac{x}{\color{blue}{z \cdot \left(1 - -0.16666666666666666 \cdot {y}^{2}\right)}} \]

    2. unpow2 68.0%

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

    3. cancel-sign-sub-inv 68.0%

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

    4. metadata-eval 68.0%

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

    5. unpow2 68.0%

       \[\leadsto \frac{x}{z \cdot \left(1 + 0.16666666666666666 \cdot \color{blue}{{y}^{2}}\right)} \]

    6. distribute-lft-in 68.0%

       \[\leadsto \frac{x}{\color{blue}{z \cdot 1 + z \cdot \left(0.16666666666666666 \cdot {y}^{2}\right)}} \]

    7. *-rgt-identity 68.0%

       \[\leadsto \frac{x}{\color{blue}{z} + z \cdot \left(0.16666666666666666 \cdot {y}^{2}\right)} \]

    8. unpow2 68.0%

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

12. Simplified 68.0%

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

13. Final simplification 68.0%

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

Alternative 8: 59.4% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8e-6) (/ x z) (* (/ y z) (/ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8e-6) {
		tmp = x / z;
	} else {
		tmp = (y / z) * (x / 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 <= 8d-6) then
        tmp = x / z
    else
        tmp = (y / z) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 8e-6) {
		tmp = x / z;
	} else {
		tmp = (y / z) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 8e-6:
		tmp = x / z
	else:
		tmp = (y / z) * (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8e-6)
		tmp = Float64(x / z);
	else
		tmp = Float64(Float64(y / z) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 8e-6)
		tmp = x / z;
	else
		tmp = (y / z) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8e-6], N[(x / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.99999999999999964e-6

   1. Initial program 97.6%

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

   2. Taylor expanded in y around 0 72.0%

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

   if 7.99999999999999964e-6 < y

   1. Initial program 90.1%

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

      1. associate-*r/ 90.2%

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

      2. associate-/l/ 89.2%

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

      3. *-commutative 89.2%

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

      4. times-frac 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in y around 0 33.3%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 9: 58.3% 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 95.7%

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

2. Taylor expanded in y around 0 60.0%

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

3. Final simplification 60.0%

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

Developer target: 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 2023194 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< z -4.2173720203427147e-29) (/ (* x (/ 1.0 (/ y (sin y)))) z) (if (< z 4.446702369113811e+64) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1.0 (/ y (sin y)))) z)))

  (/ (* x (/ (sin y) y)) z))