Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, H

Percentage Accurate: 95.8% → 97.1%

Time: 13.3s

Alternatives: 13

Speedup: 0.8×

• 28.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq 2 \cdot 10^{-195}:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot \frac{1}{z}\right) \cdot \left(y - \frac{t}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \left(3 \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 3.0) 2e-195)
   (+ x (* (* -0.3333333333333333 (/ 1.0 z)) (- y (/ t y))))
   (+ (- x (/ y (* z 3.0))) (/ t (* z (* 3.0 y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= 2e-195) {
		tmp = x + ((-0.3333333333333333 * (1.0 / z)) * (y - (t / y)));
	} else {
		tmp = (x - (y / (z * 3.0))) + (t / (z * (3.0 * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * 3.0d0) <= 2d-195) then
        tmp = x + (((-0.3333333333333333d0) * (1.0d0 / z)) * (y - (t / y)))
    else
        tmp = (x - (y / (z * 3.0d0))) + (t / (z * (3.0d0 * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= 2e-195) {
		tmp = x + ((-0.3333333333333333 * (1.0 / z)) * (y - (t / y)));
	} else {
		tmp = (x - (y / (z * 3.0))) + (t / (z * (3.0 * y)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * 3.0) <= 2e-195:
		tmp = x + ((-0.3333333333333333 * (1.0 / z)) * (y - (t / y)))
	else:
		tmp = (x - (y / (z * 3.0))) + (t / (z * (3.0 * y)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 3.0) <= 2e-195)
		tmp = Float64(x + Float64(Float64(-0.3333333333333333 * Float64(1.0 / z)) * Float64(y - Float64(t / y))));
	else
		tmp = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(z * Float64(3.0 * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * 3.0) <= 2e-195)
		tmp = x + ((-0.3333333333333333 * (1.0 / z)) * (y - (t / y)));
	else
		tmp = (x - (y / (z * 3.0))) + (t / (z * (3.0 * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 3.0), $MachinePrecision], 2e-195], N[(x + N[(N[(-0.3333333333333333 * N[(1.0 / z), $MachinePrecision]), $MachinePrecision] * N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(z * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z 3) < 2.0000000000000002e-195

   1. Initial program 91.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. Simplified 98.6%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
   4. Step-by-step derivation

      1. div-inv 98.6%

         \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   5. Applied egg-rr 98.6%

      \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   if 2.0000000000000002e-195 < (*.f64 z 3)

   1. Initial program 99.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   2. Taylor expanded in z around 0 99.8%

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

      1. *-commutative 99.8%

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(y \cdot z\right) \cdot 3}} \]

      2. *-commutative 99.8%

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{\left(z \cdot y\right)} \cdot 3} \]

      3. associate-*l* 99.8%

         \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{z \cdot \left(y \cdot 3\right)}} \]

   4. Simplified 99.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq 2 \cdot 10^{-195}:\\ \;\;\;\;x + \left(-0.3333333333333333 \cdot \frac{1}{z}\right) \cdot \left(y - \frac{t}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \left(3 \cdot y\right)}\\ \end{array} \]

Alternative 2: 91.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+22}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-28}:\\ \;\;\;\;x + \frac{\frac{-0.3333333333333333 \cdot t}{z}}{-y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -6.2e+22)
   (+ x (/ (/ y -3.0) z))
   (if (<= y 7.2e-28)
     (+ x (/ (/ (* -0.3333333333333333 t) z) (- y)))
     (+ x (/ (/ y z) -3.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.2e+22) {
		tmp = x + ((y / -3.0) / z);
	} else if (y <= 7.2e-28) {
		tmp = x + (((-0.3333333333333333 * t) / z) / -y);
	} else {
		tmp = x + ((y / z) / -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-6.2d+22)) then
        tmp = x + ((y / (-3.0d0)) / z)
    else if (y <= 7.2d-28) then
        tmp = x + ((((-0.3333333333333333d0) * t) / z) / -y)
    else
        tmp = x + ((y / z) / (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.2e+22) {
		tmp = x + ((y / -3.0) / z);
	} else if (y <= 7.2e-28) {
		tmp = x + (((-0.3333333333333333 * t) / z) / -y);
	} else {
		tmp = x + ((y / z) / -3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -6.2e+22:
		tmp = x + ((y / -3.0) / z)
	elif y <= 7.2e-28:
		tmp = x + (((-0.3333333333333333 * t) / z) / -y)
	else:
		tmp = x + ((y / z) / -3.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -6.2e+22)
		tmp = Float64(x + Float64(Float64(y / -3.0) / z));
	elseif (y <= 7.2e-28)
		tmp = Float64(x + Float64(Float64(Float64(-0.3333333333333333 * t) / z) / Float64(-y)));
	else
		tmp = Float64(x + Float64(Float64(y / z) / -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -6.2e+22)
		tmp = x + ((y / -3.0) / z);
	elseif (y <= 7.2e-28)
		tmp = x + (((-0.3333333333333333 * t) / z) / -y);
	else
		tmp = x + ((y / z) / -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -6.2e+22], N[(x + N[(N[(y / -3.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e-28], N[(x + N[(N[(N[(-0.3333333333333333 * t), $MachinePrecision] / z), $MachinePrecision] / (-y)), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y / z), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.2000000000000004e22

   1. Initial program 98.1%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
   4. Step-by-step derivation

      1. div-inv 99.7%

         \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   5. Applied egg-rr 99.7%

      \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   6. Taylor expanded in y around inf 90.0%

      \[\leadsto x + \left(-0.3333333333333333 \cdot \frac{1}{z}\right) \cdot \color{blue}{y} \]
   7. Step-by-step derivation

      1. associate-*l* 90.0%

         \[\leadsto x + \color{blue}{-0.3333333333333333 \cdot \left(\frac{1}{z} \cdot y\right)} \]

      2. associate-*l/ 90.0%

         \[\leadsto x + -0.3333333333333333 \cdot \color{blue}{\frac{1 \cdot y}{z}} \]

      3. *-un-lft-identity 90.0%

         \[\leadsto x + -0.3333333333333333 \cdot \frac{\color{blue}{y}}{z} \]

      4. metadata-eval 90.0%

         \[\leadsto x + \color{blue}{\frac{1}{-3}} \cdot \frac{y}{z} \]

      5. times-frac 90.2%

         \[\leadsto x + \color{blue}{\frac{1 \cdot y}{-3 \cdot z}} \]

      6. *-un-lft-identity 90.2%

         \[\leadsto x + \frac{\color{blue}{y}}{-3 \cdot z} \]

      7. associate-/r* 90.2%

         \[\leadsto x + \color{blue}{\frac{\frac{y}{-3}}{z}} \]

   8. Applied egg-rr 90.2%

      \[\leadsto x + \color{blue}{\frac{\frac{y}{-3}}{z}} \]

   if -6.2000000000000004e22 < y < 7.1999999999999997e-28

   1. Initial program 90.1%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. Simplified 94.3%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]

   4. Taylor expanded in y around 0 87.0%

      \[\leadsto x + \color{blue}{0.3333333333333333 \cdot \frac{t}{y \cdot z}} \]
   5. Step-by-step derivation

      1. associate-*r/ 87.0%

         \[\leadsto x + \color{blue}{\frac{0.3333333333333333 \cdot t}{y \cdot z}} \]

      2. frac-2neg 87.0%

         \[\leadsto x + \color{blue}{\frac{-0.3333333333333333 \cdot t}{-y \cdot z}} \]

      3. *-commutative 87.0%

         \[\leadsto x + \frac{-0.3333333333333333 \cdot t}{-\color{blue}{z \cdot y}} \]

      4. distribute-rgt-neg-in 87.0%

         \[\leadsto x + \frac{-0.3333333333333333 \cdot t}{\color{blue}{z \cdot \left(-y\right)}} \]

   6. Applied egg-rr 87.0%

      \[\leadsto x + \color{blue}{\frac{-0.3333333333333333 \cdot t}{z \cdot \left(-y\right)}} \]
   7. Step-by-step derivation

      1. associate-/r* 93.9%

         \[\leadsto x + \color{blue}{\frac{\frac{-0.3333333333333333 \cdot t}{z}}{-y}} \]

      2. distribute-lft-neg-in 93.9%

         \[\leadsto x + \frac{\frac{\color{blue}{\left(-0.3333333333333333\right) \cdot t}}{z}}{-y} \]

      3. metadata-eval 93.9%

         \[\leadsto x + \frac{\frac{\color{blue}{-0.3333333333333333} \cdot t}{z}}{-y} \]

   8. Simplified 93.9%

      \[\leadsto x + \color{blue}{\frac{\frac{-0.3333333333333333 \cdot t}{z}}{-y}} \]

   if 7.1999999999999997e-28 < y

   1. Initial program 99.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
   4. Step-by-step derivation

      1. div-inv 99.7%

         \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   5. Applied egg-rr 99.7%

      \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   6. Taylor expanded in y around inf 88.4%

      \[\leadsto x + \left(-0.3333333333333333 \cdot \frac{1}{z}\right) \cdot \color{blue}{y} \]
   7. Step-by-step derivation

      1. *-commutative 88.4%

         \[\leadsto x + \color{blue}{y \cdot \left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \]

      2. *-commutative 88.4%

         \[\leadsto x + y \cdot \color{blue}{\left(\frac{1}{z} \cdot -0.3333333333333333\right)} \]

      3. associate-*r* 88.3%

         \[\leadsto x + \color{blue}{\left(y \cdot \frac{1}{z}\right) \cdot -0.3333333333333333} \]

      4. div-inv 88.4%

         \[\leadsto x + \color{blue}{\frac{y}{z}} \cdot -0.3333333333333333 \]

      5. metadata-eval 88.4%

         \[\leadsto x + \frac{y}{z} \cdot \color{blue}{\frac{1}{-3}} \]

      6. div-inv 88.5%

         \[\leadsto x + \color{blue}{\frac{\frac{y}{z}}{-3}} \]

   8. Applied egg-rr 88.5%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+22}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-28}:\\ \;\;\;\;x + \frac{\frac{-0.3333333333333333 \cdot t}{z}}{-y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \end{array} \]

Alternative 3: 89.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+21}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-29}:\\ \;\;\;\;x + 0.3333333333333333 \cdot \frac{t}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -7.5e+21)
   (+ x (/ (/ y -3.0) z))
   (if (<= y 9.6e-29)
     (+ x (* 0.3333333333333333 (/ t (* z y))))
     (+ x (/ (/ y z) -3.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7.5e+21) {
		tmp = x + ((y / -3.0) / z);
	} else if (y <= 9.6e-29) {
		tmp = x + (0.3333333333333333 * (t / (z * y)));
	} else {
		tmp = x + ((y / z) / -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-7.5d+21)) then
        tmp = x + ((y / (-3.0d0)) / z)
    else if (y <= 9.6d-29) then
        tmp = x + (0.3333333333333333d0 * (t / (z * y)))
    else
        tmp = x + ((y / z) / (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7.5e+21) {
		tmp = x + ((y / -3.0) / z);
	} else if (y <= 9.6e-29) {
		tmp = x + (0.3333333333333333 * (t / (z * y)));
	} else {
		tmp = x + ((y / z) / -3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -7.5e+21:
		tmp = x + ((y / -3.0) / z)
	elif y <= 9.6e-29:
		tmp = x + (0.3333333333333333 * (t / (z * y)))
	else:
		tmp = x + ((y / z) / -3.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7.5e+21)
		tmp = Float64(x + Float64(Float64(y / -3.0) / z));
	elseif (y <= 9.6e-29)
		tmp = Float64(x + Float64(0.3333333333333333 * Float64(t / Float64(z * y))));
	else
		tmp = Float64(x + Float64(Float64(y / z) / -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -7.5e+21)
		tmp = x + ((y / -3.0) / z);
	elseif (y <= 9.6e-29)
		tmp = x + (0.3333333333333333 * (t / (z * y)));
	else
		tmp = x + ((y / z) / -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -7.5e+21], N[(x + N[(N[(y / -3.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.6e-29], N[(x + N[(0.3333333333333333 * N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y / z), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.5e21

   1. Initial program 98.1%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
   4. Step-by-step derivation

      1. div-inv 99.7%

         \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   5. Applied egg-rr 99.7%

      \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   6. Taylor expanded in y around inf 90.0%

      \[\leadsto x + \left(-0.3333333333333333 \cdot \frac{1}{z}\right) \cdot \color{blue}{y} \]
   7. Step-by-step derivation

      1. associate-*l* 90.0%

         \[\leadsto x + \color{blue}{-0.3333333333333333 \cdot \left(\frac{1}{z} \cdot y\right)} \]

      2. associate-*l/ 90.0%

         \[\leadsto x + -0.3333333333333333 \cdot \color{blue}{\frac{1 \cdot y}{z}} \]

      3. *-un-lft-identity 90.0%

         \[\leadsto x + -0.3333333333333333 \cdot \frac{\color{blue}{y}}{z} \]

      4. metadata-eval 90.0%

         \[\leadsto x + \color{blue}{\frac{1}{-3}} \cdot \frac{y}{z} \]

      5. times-frac 90.2%

         \[\leadsto x + \color{blue}{\frac{1 \cdot y}{-3 \cdot z}} \]

      6. *-un-lft-identity 90.2%

         \[\leadsto x + \frac{\color{blue}{y}}{-3 \cdot z} \]

      7. associate-/r* 90.2%

         \[\leadsto x + \color{blue}{\frac{\frac{y}{-3}}{z}} \]

   8. Applied egg-rr 90.2%

      \[\leadsto x + \color{blue}{\frac{\frac{y}{-3}}{z}} \]

   if -7.5e21 < y < 9.59999999999999968e-29

   1. Initial program 90.1%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. Simplified 94.3%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]

   4. Taylor expanded in y around 0 87.0%

      \[\leadsto x + \color{blue}{0.3333333333333333 \cdot \frac{t}{y \cdot z}} \]

   if 9.59999999999999968e-29 < y

   1. Initial program 99.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
   4. Step-by-step derivation

      1. div-inv 99.7%

         \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   5. Applied egg-rr 99.7%

      \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   6. Taylor expanded in y around inf 88.4%

      \[\leadsto x + \left(-0.3333333333333333 \cdot \frac{1}{z}\right) \cdot \color{blue}{y} \]
   7. Step-by-step derivation

      1. *-commutative 88.4%

         \[\leadsto x + \color{blue}{y \cdot \left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \]

      2. *-commutative 88.4%

         \[\leadsto x + y \cdot \color{blue}{\left(\frac{1}{z} \cdot -0.3333333333333333\right)} \]

      3. associate-*r* 88.3%

         \[\leadsto x + \color{blue}{\left(y \cdot \frac{1}{z}\right) \cdot -0.3333333333333333} \]

      4. div-inv 88.4%

         \[\leadsto x + \color{blue}{\frac{y}{z}} \cdot -0.3333333333333333 \]

      5. metadata-eval 88.4%

         \[\leadsto x + \frac{y}{z} \cdot \color{blue}{\frac{1}{-3}} \]

      6. div-inv 88.5%

         \[\leadsto x + \color{blue}{\frac{\frac{y}{z}}{-3}} \]

   8. Applied egg-rr 88.5%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+21}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-29}:\\ \;\;\;\;x + 0.3333333333333333 \cdot \frac{t}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \end{array} \]

Alternative 4: 88.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+22}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-28}:\\ \;\;\;\;x + \frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -6.2e+22)
   (+ x (/ (/ y -3.0) z))
   (if (<= y 9e-28)
     (+ x (* (/ t y) (/ 0.3333333333333333 z)))
     (+ x (/ (/ y z) -3.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.2e+22) {
		tmp = x + ((y / -3.0) / z);
	} else if (y <= 9e-28) {
		tmp = x + ((t / y) * (0.3333333333333333 / z));
	} else {
		tmp = x + ((y / z) / -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-6.2d+22)) then
        tmp = x + ((y / (-3.0d0)) / z)
    else if (y <= 9d-28) then
        tmp = x + ((t / y) * (0.3333333333333333d0 / z))
    else
        tmp = x + ((y / z) / (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.2e+22) {
		tmp = x + ((y / -3.0) / z);
	} else if (y <= 9e-28) {
		tmp = x + ((t / y) * (0.3333333333333333 / z));
	} else {
		tmp = x + ((y / z) / -3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -6.2e+22:
		tmp = x + ((y / -3.0) / z)
	elif y <= 9e-28:
		tmp = x + ((t / y) * (0.3333333333333333 / z))
	else:
		tmp = x + ((y / z) / -3.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -6.2e+22)
		tmp = Float64(x + Float64(Float64(y / -3.0) / z));
	elseif (y <= 9e-28)
		tmp = Float64(x + Float64(Float64(t / y) * Float64(0.3333333333333333 / z)));
	else
		tmp = Float64(x + Float64(Float64(y / z) / -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -6.2e+22)
		tmp = x + ((y / -3.0) / z);
	elseif (y <= 9e-28)
		tmp = x + ((t / y) * (0.3333333333333333 / z));
	else
		tmp = x + ((y / z) / -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -6.2e+22], N[(x + N[(N[(y / -3.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e-28], N[(x + N[(N[(t / y), $MachinePrecision] * N[(0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y / z), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.2000000000000004e22

   1. Initial program 98.1%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
   4. Step-by-step derivation

      1. div-inv 99.7%

         \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   5. Applied egg-rr 99.7%

      \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   6. Taylor expanded in y around inf 90.0%

      \[\leadsto x + \left(-0.3333333333333333 \cdot \frac{1}{z}\right) \cdot \color{blue}{y} \]
   7. Step-by-step derivation

      1. associate-*l* 90.0%

         \[\leadsto x + \color{blue}{-0.3333333333333333 \cdot \left(\frac{1}{z} \cdot y\right)} \]

      2. associate-*l/ 90.0%

         \[\leadsto x + -0.3333333333333333 \cdot \color{blue}{\frac{1 \cdot y}{z}} \]

      3. *-un-lft-identity 90.0%

         \[\leadsto x + -0.3333333333333333 \cdot \frac{\color{blue}{y}}{z} \]

      4. metadata-eval 90.0%

         \[\leadsto x + \color{blue}{\frac{1}{-3}} \cdot \frac{y}{z} \]

      5. times-frac 90.2%

         \[\leadsto x + \color{blue}{\frac{1 \cdot y}{-3 \cdot z}} \]

      6. *-un-lft-identity 90.2%

         \[\leadsto x + \frac{\color{blue}{y}}{-3 \cdot z} \]

      7. associate-/r* 90.2%

         \[\leadsto x + \color{blue}{\frac{\frac{y}{-3}}{z}} \]

   8. Applied egg-rr 90.2%

      \[\leadsto x + \color{blue}{\frac{\frac{y}{-3}}{z}} \]

   if -6.2000000000000004e22 < y < 8.9999999999999996e-28

   1. Initial program 90.1%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. Simplified 94.3%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]

   4. Taylor expanded in y around 0 87.0%

      \[\leadsto x + \color{blue}{0.3333333333333333 \cdot \frac{t}{y \cdot z}} \]
   5. Step-by-step derivation

      1. associate-/r* 89.5%

         \[\leadsto x + 0.3333333333333333 \cdot \color{blue}{\frac{\frac{t}{y}}{z}} \]

      2. associate-*r/ 89.5%

         \[\leadsto x + \color{blue}{\frac{0.3333333333333333 \cdot \frac{t}{y}}{z}} \]

      3. *-commutative 89.5%

         \[\leadsto x + \frac{\color{blue}{\frac{t}{y} \cdot 0.3333333333333333}}{z} \]

      4. associate-*r/ 89.5%

         \[\leadsto x + \color{blue}{\frac{t}{y} \cdot \frac{0.3333333333333333}{z}} \]

   6. Simplified 89.5%

      \[\leadsto x + \color{blue}{\frac{t}{y} \cdot \frac{0.3333333333333333}{z}} \]

   if 8.9999999999999996e-28 < y

   1. Initial program 99.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
   4. Step-by-step derivation

      1. div-inv 99.7%

         \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   5. Applied egg-rr 99.7%

      \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   6. Taylor expanded in y around inf 88.4%

      \[\leadsto x + \left(-0.3333333333333333 \cdot \frac{1}{z}\right) \cdot \color{blue}{y} \]
   7. Step-by-step derivation

      1. *-commutative 88.4%

         \[\leadsto x + \color{blue}{y \cdot \left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \]

      2. *-commutative 88.4%

         \[\leadsto x + y \cdot \color{blue}{\left(\frac{1}{z} \cdot -0.3333333333333333\right)} \]

      3. associate-*r* 88.3%

         \[\leadsto x + \color{blue}{\left(y \cdot \frac{1}{z}\right) \cdot -0.3333333333333333} \]

      4. div-inv 88.4%

         \[\leadsto x + \color{blue}{\frac{y}{z}} \cdot -0.3333333333333333 \]

      5. metadata-eval 88.4%

         \[\leadsto x + \frac{y}{z} \cdot \color{blue}{\frac{1}{-3}} \]

      6. div-inv 88.5%

         \[\leadsto x + \color{blue}{\frac{\frac{y}{z}}{-3}} \]

   8. Applied egg-rr 88.5%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+22}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-28}:\\ \;\;\;\;x + \frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \end{array} \]

Alternative 5: 88.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+22}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-28}:\\ \;\;\;\;x + \frac{0.3333333333333333}{z \cdot \frac{y}{t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.8e+22)
   (+ x (/ (/ y -3.0) z))
   (if (<= y 9e-28)
     (+ x (/ 0.3333333333333333 (* z (/ y t))))
     (+ x (/ (/ y z) -3.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.8e+22) {
		tmp = x + ((y / -3.0) / z);
	} else if (y <= 9e-28) {
		tmp = x + (0.3333333333333333 / (z * (y / t)));
	} else {
		tmp = x + ((y / z) / -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-2.8d+22)) then
        tmp = x + ((y / (-3.0d0)) / z)
    else if (y <= 9d-28) then
        tmp = x + (0.3333333333333333d0 / (z * (y / t)))
    else
        tmp = x + ((y / z) / (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.8e+22) {
		tmp = x + ((y / -3.0) / z);
	} else if (y <= 9e-28) {
		tmp = x + (0.3333333333333333 / (z * (y / t)));
	} else {
		tmp = x + ((y / z) / -3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2.8e+22:
		tmp = x + ((y / -3.0) / z)
	elif y <= 9e-28:
		tmp = x + (0.3333333333333333 / (z * (y / t)))
	else:
		tmp = x + ((y / z) / -3.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.8e+22)
		tmp = Float64(x + Float64(Float64(y / -3.0) / z));
	elseif (y <= 9e-28)
		tmp = Float64(x + Float64(0.3333333333333333 / Float64(z * Float64(y / t))));
	else
		tmp = Float64(x + Float64(Float64(y / z) / -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2.8e+22)
		tmp = x + ((y / -3.0) / z);
	elseif (y <= 9e-28)
		tmp = x + (0.3333333333333333 / (z * (y / t)));
	else
		tmp = x + ((y / z) / -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.8e+22], N[(x + N[(N[(y / -3.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e-28], N[(x + N[(0.3333333333333333 / N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y / z), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.8e22

   1. Initial program 98.1%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
   4. Step-by-step derivation

      1. div-inv 99.7%

         \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   5. Applied egg-rr 99.7%

      \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   6. Taylor expanded in y around inf 90.0%

      \[\leadsto x + \left(-0.3333333333333333 \cdot \frac{1}{z}\right) \cdot \color{blue}{y} \]
   7. Step-by-step derivation

      1. associate-*l* 90.0%

         \[\leadsto x + \color{blue}{-0.3333333333333333 \cdot \left(\frac{1}{z} \cdot y\right)} \]

      2. associate-*l/ 90.0%

         \[\leadsto x + -0.3333333333333333 \cdot \color{blue}{\frac{1 \cdot y}{z}} \]

      3. *-un-lft-identity 90.0%

         \[\leadsto x + -0.3333333333333333 \cdot \frac{\color{blue}{y}}{z} \]

      4. metadata-eval 90.0%

         \[\leadsto x + \color{blue}{\frac{1}{-3}} \cdot \frac{y}{z} \]

      5. times-frac 90.2%

         \[\leadsto x + \color{blue}{\frac{1 \cdot y}{-3 \cdot z}} \]

      6. *-un-lft-identity 90.2%

         \[\leadsto x + \frac{\color{blue}{y}}{-3 \cdot z} \]

      7. associate-/r* 90.2%

         \[\leadsto x + \color{blue}{\frac{\frac{y}{-3}}{z}} \]

   8. Applied egg-rr 90.2%

      \[\leadsto x + \color{blue}{\frac{\frac{y}{-3}}{z}} \]

   if -2.8e22 < y < 8.9999999999999996e-28

   1. Initial program 90.1%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. Simplified 94.3%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]

   4. Taylor expanded in y around 0 87.0%

      \[\leadsto x + \color{blue}{0.3333333333333333 \cdot \frac{t}{y \cdot z}} \]
   5. Step-by-step derivation

      1. clear-num 86.9%

         \[\leadsto x + 0.3333333333333333 \cdot \color{blue}{\frac{1}{\frac{y \cdot z}{t}}} \]

      2. un-div-inv 87.0%

         \[\leadsto x + \color{blue}{\frac{0.3333333333333333}{\frac{y \cdot z}{t}}} \]

      3. *-commutative 87.0%

         \[\leadsto x + \frac{0.3333333333333333}{\frac{\color{blue}{z \cdot y}}{t}} \]

      4. *-un-lft-identity 87.0%

         \[\leadsto x + \frac{0.3333333333333333}{\frac{z \cdot y}{\color{blue}{1 \cdot t}}} \]

      5. times-frac 90.0%

         \[\leadsto x + \frac{0.3333333333333333}{\color{blue}{\frac{z}{1} \cdot \frac{y}{t}}} \]

      6. /-rgt-identity 90.0%

         \[\leadsto x + \frac{0.3333333333333333}{\color{blue}{z} \cdot \frac{y}{t}} \]

   6. Applied egg-rr 90.0%

      \[\leadsto x + \color{blue}{\frac{0.3333333333333333}{z \cdot \frac{y}{t}}} \]

   if 8.9999999999999996e-28 < y

   1. Initial program 99.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
   4. Step-by-step derivation

      1. div-inv 99.7%

         \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   5. Applied egg-rr 99.7%

      \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

   6. Taylor expanded in y around inf 88.4%

      \[\leadsto x + \left(-0.3333333333333333 \cdot \frac{1}{z}\right) \cdot \color{blue}{y} \]
   7. Step-by-step derivation

      1. *-commutative 88.4%

         \[\leadsto x + \color{blue}{y \cdot \left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \]

      2. *-commutative 88.4%

         \[\leadsto x + y \cdot \color{blue}{\left(\frac{1}{z} \cdot -0.3333333333333333\right)} \]

      3. associate-*r* 88.3%

         \[\leadsto x + \color{blue}{\left(y \cdot \frac{1}{z}\right) \cdot -0.3333333333333333} \]

      4. div-inv 88.4%

         \[\leadsto x + \color{blue}{\frac{y}{z}} \cdot -0.3333333333333333 \]

      5. metadata-eval 88.4%

         \[\leadsto x + \frac{y}{z} \cdot \color{blue}{\frac{1}{-3}} \]

      6. div-inv 88.5%

         \[\leadsto x + \color{blue}{\frac{\frac{y}{z}}{-3}} \]

   8. Applied egg-rr 88.5%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+22}:\\ \;\;\;\;x + \frac{\frac{y}{-3}}{z}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-28}:\\ \;\;\;\;x + \frac{0.3333333333333333}{z \cdot \frac{y}{t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y}{z}}{-3}\\ \end{array} \]

Alternative 6: 95.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ x + \left(-0.3333333333333333 \cdot \frac{1}{z}\right) \cdot \left(y - \frac{t}{y}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ x (* (* -0.3333333333333333 (/ 1.0 z)) (- y (/ t y)))))

C

double code(double x, double y, double z, double t) {
	return x + ((-0.3333333333333333 * (1.0 / z)) * (y - (t / y)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + (((-0.3333333333333333d0) * (1.0d0 / z)) * (y - (t / y)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + ((-0.3333333333333333 * (1.0 / z)) * (y - (t / y)));
}

Python

def code(x, y, z, t):
	return x + ((-0.3333333333333333 * (1.0 / z)) * (y - (t / y)))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(-0.3333333333333333 * Float64(1.0 / z)) * Float64(y - Float64(t / y))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + ((-0.3333333333333333 * (1.0 / z)) * (y - (t / y)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(-0.3333333333333333 * N[(1.0 / z), $MachinePrecision]), $MachinePrecision] * N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.9%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

3. Simplified 97.2%

   \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
4. Step-by-step derivation

   1. div-inv 97.2%

      \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

5. Applied egg-rr 97.2%

   \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

6. Final simplification 97.2%

   \[\leadsto x + \left(-0.3333333333333333 \cdot \frac{1}{z}\right) \cdot \left(y - \frac{t}{y}\right) \]

Alternative 7: 95.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ x + -0.3333333333333333 \cdot \frac{y - \frac{t}{y}}{z} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ x (* -0.3333333333333333 (/ (- y (/ t y)) z))))

C

double code(double x, double y, double z, double t) {
	return x + (-0.3333333333333333 * ((y - (t / y)) / z));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((-0.3333333333333333d0) * ((y - (t / y)) / z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + (-0.3333333333333333 * ((y - (t / y)) / z));
}

Python

def code(x, y, z, t):
	return x + (-0.3333333333333333 * ((y - (t / y)) / z))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(-0.3333333333333333 * Float64(Float64(y - Float64(t / y)) / z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + (-0.3333333333333333 * ((y - (t / y)) / z));
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(-0.3333333333333333 * N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.9%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

3. Simplified 97.2%

   \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]

4. Taylor expanded in z around 0 97.2%

   \[\leadsto x + \color{blue}{-0.3333333333333333 \cdot \frac{y - \frac{t}{y}}{z}} \]

5. Final simplification 97.2%

   \[\leadsto x + -0.3333333333333333 \cdot \frac{y - \frac{t}{y}}{z} \]

Alternative 8: 95.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ x (* (- y (/ t y)) (/ -0.3333333333333333 z))))

C

double code(double x, double y, double z, double t) {
	return x + ((y - (t / y)) * (-0.3333333333333333 / z));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y - (t / y)) * ((-0.3333333333333333d0) / z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + ((y - (t / y)) * (-0.3333333333333333 / z));
}

Python

def code(x, y, z, t):
	return x + ((y - (t / y)) * (-0.3333333333333333 / z))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y - Float64(t / y)) * Float64(-0.3333333333333333 / z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + ((y - (t / y)) * (-0.3333333333333333 / z));
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.9%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

3. Simplified 97.2%

   \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]

4. Final simplification 97.2%

   \[\leadsto x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z} \]

Alternative 9: 64.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ x + y \cdot \frac{-0.3333333333333333}{z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ x (* y (/ -0.3333333333333333 z))))

C

double code(double x, double y, double z, double t) {
	return x + (y * (-0.3333333333333333 / z));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + (y * ((-0.3333333333333333d0) / z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + (y * (-0.3333333333333333 / z));
}

Python

def code(x, y, z, t):
	return x + (y * (-0.3333333333333333 / z))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(y * Float64(-0.3333333333333333 / z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + (y * (-0.3333333333333333 / z));
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(y * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.9%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

3. Simplified 97.2%

   \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]

4. Taylor expanded in y around inf 64.9%

   \[\leadsto x + \frac{-0.3333333333333333}{z} \cdot \color{blue}{y} \]

5. Final simplification 64.9%

   \[\leadsto x + y \cdot \frac{-0.3333333333333333}{z} \]

Alternative 10: 64.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ x + \frac{-0.3333333333333333}{\frac{z}{y}} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ x (/ -0.3333333333333333 (/ z y))))

C

double code(double x, double y, double z, double t) {
	return x + (-0.3333333333333333 / (z / y));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((-0.3333333333333333d0) / (z / y))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + (-0.3333333333333333 / (z / y));
}

Python

def code(x, y, z, t):
	return x + (-0.3333333333333333 / (z / y))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(-0.3333333333333333 / Float64(z / y)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + (-0.3333333333333333 / (z / y));
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(-0.3333333333333333 / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.9%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

3. Simplified 97.2%

   \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
4. Step-by-step derivation

   1. div-inv 97.2%

      \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

5. Applied egg-rr 97.2%

   \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

6. Taylor expanded in y around inf 65.0%

   \[\leadsto x + \left(-0.3333333333333333 \cdot \frac{1}{z}\right) \cdot \color{blue}{y} \]
7. Step-by-step derivation

   1. un-div-inv 64.9%

      \[\leadsto x + \color{blue}{\frac{-0.3333333333333333}{z}} \cdot y \]

   2. associate-/r/ 65.0%

      \[\leadsto x + \color{blue}{\frac{-0.3333333333333333}{\frac{z}{y}}} \]

8. Applied egg-rr 65.0%

   \[\leadsto x + \color{blue}{\frac{-0.3333333333333333}{\frac{z}{y}}} \]

9. Final simplification 65.0%

   \[\leadsto x + \frac{-0.3333333333333333}{\frac{z}{y}} \]

Alternative 11: 64.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ x + \frac{y}{z \cdot -3} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ x (/ y (* z -3.0))))

C

double code(double x, double y, double z, double t) {
	return x + (y / (z * -3.0));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + (y / (z * (-3.0d0)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + (y / (z * -3.0));
}

Python

def code(x, y, z, t):
	return x + (y / (z * -3.0))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(y / Float64(z * -3.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + (y / (z * -3.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(y / N[(z * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.9%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

3. Simplified 97.2%

   \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]

4. Taylor expanded in y around inf 64.9%

   \[\leadsto x + \frac{-0.3333333333333333}{z} \cdot \color{blue}{y} \]
5. Step-by-step derivation

   1. expm1-log1p-u 45.2%

      \[\leadsto x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.3333333333333333}{z} \cdot y\right)\right)} \]

   2. expm1-udef 42.0%

      \[\leadsto x + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.3333333333333333}{z} \cdot y\right)} - 1\right)} \]

   3. *-commutative 42.0%

      \[\leadsto x + \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \frac{-0.3333333333333333}{z}}\right)} - 1\right) \]

   4. clear-num 42.0%

      \[\leadsto x + \left(e^{\mathsf{log1p}\left(y \cdot \color{blue}{\frac{1}{\frac{z}{-0.3333333333333333}}}\right)} - 1\right) \]

   5. un-div-inv 42.0%

      \[\leadsto x + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{y}{\frac{z}{-0.3333333333333333}}}\right)} - 1\right) \]

   6. div-inv 42.0%

      \[\leadsto x + \left(e^{\mathsf{log1p}\left(\frac{y}{\color{blue}{z \cdot \frac{1}{-0.3333333333333333}}}\right)} - 1\right) \]

   7. metadata-eval 42.0%

      \[\leadsto x + \left(e^{\mathsf{log1p}\left(\frac{y}{z \cdot \color{blue}{-3}}\right)} - 1\right) \]

6. Applied egg-rr 42.0%

   \[\leadsto x + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y}{z \cdot -3}\right)} - 1\right)} \]
7. Step-by-step derivation

   1. expm1-def 45.2%

      \[\leadsto x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{z \cdot -3}\right)\right)} \]

   2. expm1-log1p 65.0%

      \[\leadsto x + \color{blue}{\frac{y}{z \cdot -3}} \]

8. Simplified 65.0%

   \[\leadsto x + \color{blue}{\frac{y}{z \cdot -3}} \]

9. Final simplification 65.0%

   \[\leadsto x + \frac{y}{z \cdot -3} \]

Alternative 12: 64.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ x + \frac{\frac{y}{-3}}{z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ x (/ (/ y -3.0) z)))

C

double code(double x, double y, double z, double t) {
	return x + ((y / -3.0) / z);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y / (-3.0d0)) / z)
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + ((y / -3.0) / z);
}

Python

def code(x, y, z, t):
	return x + ((y / -3.0) / z)

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y / -3.0) / z))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + ((y / -3.0) / z);
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(y / -3.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.9%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

3. Simplified 97.2%

   \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
4. Step-by-step derivation

   1. div-inv 97.2%

      \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

5. Applied egg-rr 97.2%

   \[\leadsto x + \color{blue}{\left(-0.3333333333333333 \cdot \frac{1}{z}\right)} \cdot \left(y - \frac{t}{y}\right) \]

6. Taylor expanded in y around inf 65.0%

   \[\leadsto x + \left(-0.3333333333333333 \cdot \frac{1}{z}\right) \cdot \color{blue}{y} \]
7. Step-by-step derivation

   1. associate-*l* 64.9%

      \[\leadsto x + \color{blue}{-0.3333333333333333 \cdot \left(\frac{1}{z} \cdot y\right)} \]

   2. associate-*l/ 65.0%

      \[\leadsto x + -0.3333333333333333 \cdot \color{blue}{\frac{1 \cdot y}{z}} \]

   3. *-un-lft-identity 65.0%

      \[\leadsto x + -0.3333333333333333 \cdot \frac{\color{blue}{y}}{z} \]

   4. metadata-eval 65.0%

      \[\leadsto x + \color{blue}{\frac{1}{-3}} \cdot \frac{y}{z} \]

   5. times-frac 65.0%

      \[\leadsto x + \color{blue}{\frac{1 \cdot y}{-3 \cdot z}} \]

   6. *-un-lft-identity 65.0%

      \[\leadsto x + \frac{\color{blue}{y}}{-3 \cdot z} \]

   7. associate-/r* 65.0%

      \[\leadsto x + \color{blue}{\frac{\frac{y}{-3}}{z}} \]

8. Applied egg-rr 65.0%

   \[\leadsto x + \color{blue}{\frac{\frac{y}{-3}}{z}} \]

9. Final simplification 65.0%

   \[\leadsto x + \frac{\frac{y}{-3}}{z} \]

Alternative 13: 30.7% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x
end function

Java

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

Python

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

Julia

function code(x, y, z, t)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 94.9%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

3. Simplified 97.2%

   \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]

4. Taylor expanded in x around inf 30.7%

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

5. Final simplification 30.7%

   \[\leadsto x \]

Developer target: 96.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y)))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + ((t / (z * 3.0d0)) / y)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y)

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(Float64(t / Float64(z * 3.0)) / y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t / N[(z * 3.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :herbie-target
  (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y))

  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))