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

Percentage Accurate: 95.5% → 95.5%

Time: 10.4s

Alternatives: 17

Speedup: 1.0×

• 27.5% 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: 95.5% 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: 95.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.1%

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

2. Final simplification 97.1%

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

Alternative 2: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.56 \cdot 10^{+69}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq -4.2:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\frac{t}{y} - y\right)}{z}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{0.3333333333333333}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.56e+69)
   (- x (/ y (* z 3.0)))
   (if (<= y -4.2)
     (/ (* 0.3333333333333333 (- (/ t y) y)) z)
     (if (<= y 1.25e-17)
       (+ x (/ (/ t (* z 3.0)) y))
       (- x (* y (/ 0.3333333333333333 z)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.56e+69) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= -4.2) {
		tmp = (0.3333333333333333 * ((t / y) - y)) / z;
	} else if (y <= 1.25e-17) {
		tmp = x + ((t / (z * 3.0)) / y);
	} else {
		tmp = x - (y * (0.3333333333333333 / z));
	}
	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 <= (-1.56d+69)) then
        tmp = x - (y / (z * 3.0d0))
    else if (y <= (-4.2d0)) then
        tmp = (0.3333333333333333d0 * ((t / y) - y)) / z
    else if (y <= 1.25d-17) then
        tmp = x + ((t / (z * 3.0d0)) / y)
    else
        tmp = x - (y * (0.3333333333333333d0 / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.56e+69) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= -4.2) {
		tmp = (0.3333333333333333 * ((t / y) - y)) / z;
	} else if (y <= 1.25e-17) {
		tmp = x + ((t / (z * 3.0)) / y);
	} else {
		tmp = x - (y * (0.3333333333333333 / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.56e+69:
		tmp = x - (y / (z * 3.0))
	elif y <= -4.2:
		tmp = (0.3333333333333333 * ((t / y) - y)) / z
	elif y <= 1.25e-17:
		tmp = x + ((t / (z * 3.0)) / y)
	else:
		tmp = x - (y * (0.3333333333333333 / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.56e+69)
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	elseif (y <= -4.2)
		tmp = Float64(Float64(0.3333333333333333 * Float64(Float64(t / y) - y)) / z);
	elseif (y <= 1.25e-17)
		tmp = Float64(x + Float64(Float64(t / Float64(z * 3.0)) / y));
	else
		tmp = Float64(x - Float64(y * Float64(0.3333333333333333 / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.56e+69)
		tmp = x - (y / (z * 3.0));
	elseif (y <= -4.2)
		tmp = (0.3333333333333333 * ((t / y) - y)) / z;
	elseif (y <= 1.25e-17)
		tmp = x + ((t / (z * 3.0)) / y);
	else
		tmp = x - (y * (0.3333333333333333 / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.56e+69], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -4.2], N[(N[(0.3333333333333333 * N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 1.25e-17], N[(x + N[(N[(t / N[(z * 3.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.56000000000000007e69

   1. Initial program 97.8%

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

      1. associate-*l* 97.8%

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

      2. *-commutative 97.8%

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

   3. Simplified 97.8%

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

      1. associate-+l- 97.8%

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

      2. *-commutative 97.8%

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

      3. associate-*l* 97.8%

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

      4. associate-/l/ 97.8%

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

      5. sub-div 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around inf 96.2%

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

   if -1.56000000000000007e69 < y < -4.20000000000000018

   1. Initial program 99.8%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. *-commutative 99.7%

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

      4. *-un-lft-identity 99.7%

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

      5. times-frac 99.5%

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

      6. metadata-eval 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in x around 0 95.0%

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

      1. fma-neg 95.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333, \frac{t}{y \cdot z}, -0.3333333333333333 \cdot \frac{y}{z}\right)} \]

      2. *-commutative 95.0%

         \[\leadsto \mathsf{fma}\left(0.3333333333333333, \frac{t}{\color{blue}{z \cdot y}}, -0.3333333333333333 \cdot \frac{y}{z}\right) \]

      3. fma-neg 95.0%

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

      4. associate-*r/ 95.0%

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

      5. times-frac 95.0%

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

      6. associate-*l/ 95.1%

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

      7. associate-*r/ 95.3%

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

      8. div-sub 95.3%

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

      9. distribute-lft-out-- 95.3%

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

   6. Simplified 95.3%

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

   if -4.20000000000000018 < y < 1.25e-17

   1. Initial program 96.3%

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

      1. associate-*l* 96.2%

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

      2. *-commutative 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in y around 0 94.1%

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

      1. +-commutative 94.1%

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

      2. *-commutative 94.1%

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

      3. associate-/r* 89.9%

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

      4. associate-*l/ 89.9%

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

      5. associate-*r/ 89.9%

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

      6. associate-*l/ 94.6%

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

   6. Simplified 94.6%

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

   7. Taylor expanded in t around 0 94.7%

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

      1. associate-*r/ 94.6%

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

      2. *-commutative 94.6%

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

      3. /-rgt-identity 94.6%

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

      4. associate-/l* 94.7%

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

      5. metadata-eval 94.7%

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

      6. associate-/l/ 94.8%

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

   9. Simplified 94.8%

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

   if 1.25e-17 < y

   1. Initial program 97.2%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 91.5%

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

      1. neg-mul-1 91.5%

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

   6. Simplified 91.5%

      \[\leadsto x + \frac{0.3333333333333333}{z} \cdot \color{blue}{\left(-y\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.56 \cdot 10^{+69}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq -4.2:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\frac{t}{y} - y\right)}{z}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{0.3333333333333333}{z}\\ \end{array} \]

Alternative 3: 95.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / (z * (y * 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))) + (t / (z * (y * 3.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.1%

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

   1. associate-*l* 97.1%

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

   2. *-commutative 97.1%

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

3. Simplified 97.1%

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

4. Final simplification 97.1%

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

Alternative 4: 89.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-9}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 0.068:\\ \;\;\;\;x + 0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{0.3333333333333333}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -7e-9)
   (- x (/ y (* z 3.0)))
   (if (<= y 0.068)
     (+ x (* 0.3333333333333333 (/ t (* y z))))
     (- x (* y (/ 0.3333333333333333 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7e-9) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 0.068) {
		tmp = x + (0.3333333333333333 * (t / (y * z)));
	} else {
		tmp = x - (y * (0.3333333333333333 / z));
	}
	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 <= (-7d-9)) then
        tmp = x - (y / (z * 3.0d0))
    else if (y <= 0.068d0) then
        tmp = x + (0.3333333333333333d0 * (t / (y * z)))
    else
        tmp = x - (y * (0.3333333333333333d0 / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7e-9) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 0.068) {
		tmp = x + (0.3333333333333333 * (t / (y * z)));
	} else {
		tmp = x - (y * (0.3333333333333333 / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -7e-9:
		tmp = x - (y / (z * 3.0))
	elif y <= 0.068:
		tmp = x + (0.3333333333333333 * (t / (y * z)))
	else:
		tmp = x - (y * (0.3333333333333333 / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7e-9)
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	elseif (y <= 0.068)
		tmp = Float64(x + Float64(0.3333333333333333 * Float64(t / Float64(y * z))));
	else
		tmp = Float64(x - Float64(y * Float64(0.3333333333333333 / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -7e-9)
		tmp = x - (y / (z * 3.0));
	elseif (y <= 0.068)
		tmp = x + (0.3333333333333333 * (t / (y * z)));
	else
		tmp = x - (y * (0.3333333333333333 / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -7e-9], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.068], N[(x + N[(0.3333333333333333 * N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.9999999999999998e-9

   1. Initial program 98.4%

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

      1. associate-*l* 98.4%

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

      2. *-commutative 98.4%

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

   3. Simplified 98.4%

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

      1. associate-+l- 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-*l* 98.4%

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

      4. associate-/l/ 98.4%

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

      5. sub-div 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around inf 89.3%

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

   if -6.9999999999999998e-9 < y < 0.068000000000000005

   1. Initial program 96.2%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in t around inf 94.8%

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

   if 0.068000000000000005 < y

   1. Initial program 97.2%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 91.5%

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

      1. neg-mul-1 91.5%

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

   6. Simplified 91.5%

      \[\leadsto x + \frac{0.3333333333333333}{z} \cdot \color{blue}{\left(-y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-9}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 0.068:\\ \;\;\;\;x + 0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{0.3333333333333333}{z}\\ \end{array} \]

Alternative 5: 89.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-8}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 0.0112:\\ \;\;\;\;x + \frac{t}{3 \cdot \left(y \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{0.3333333333333333}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -7.5e-8)
   (- x (/ y (* z 3.0)))
   (if (<= y 0.0112)
     (+ x (/ t (* 3.0 (* y z))))
     (- x (* y (/ 0.3333333333333333 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7.5e-8) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 0.0112) {
		tmp = x + (t / (3.0 * (y * z)));
	} else {
		tmp = x - (y * (0.3333333333333333 / z));
	}
	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-8)) then
        tmp = x - (y / (z * 3.0d0))
    else if (y <= 0.0112d0) then
        tmp = x + (t / (3.0d0 * (y * z)))
    else
        tmp = x - (y * (0.3333333333333333d0 / z))
    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-8) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 0.0112) {
		tmp = x + (t / (3.0 * (y * z)));
	} else {
		tmp = x - (y * (0.3333333333333333 / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -7.5e-8:
		tmp = x - (y / (z * 3.0))
	elif y <= 0.0112:
		tmp = x + (t / (3.0 * (y * z)))
	else:
		tmp = x - (y * (0.3333333333333333 / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7.5e-8)
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	elseif (y <= 0.0112)
		tmp = Float64(x + Float64(t / Float64(3.0 * Float64(y * z))));
	else
		tmp = Float64(x - Float64(y * Float64(0.3333333333333333 / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -7.5e-8)
		tmp = x - (y / (z * 3.0));
	elseif (y <= 0.0112)
		tmp = x + (t / (3.0 * (y * z)));
	else
		tmp = x - (y * (0.3333333333333333 / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -7.5e-8], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0112], N[(x + N[(t / N[(3.0 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.4999999999999997e-8

   1. Initial program 98.4%

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

      1. associate-*l* 98.4%

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

      2. *-commutative 98.4%

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

   3. Simplified 98.4%

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

      1. associate-+l- 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-*l* 98.4%

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

      4. associate-/l/ 98.4%

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

      5. sub-div 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around inf 89.3%

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

   if -7.4999999999999997e-8 < y < 0.0111999999999999999

   1. Initial program 96.2%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in t around inf 94.8%

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

      1. associate-*r/ 94.9%

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

      2. *-commutative 94.9%

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

      3. frac-times 90.5%

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

      4. clear-num 90.4%

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

      5. frac-times 94.9%

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

      6. *-un-lft-identity 94.9%

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

      7. div-inv 95.0%

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

      8. metadata-eval 95.0%

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

      9. associate-*l* 94.9%

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

   6. Applied egg-rr 94.9%

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

   7. Taylor expanded in z around 0 94.9%

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

   if 0.0111999999999999999 < y

   1. Initial program 97.2%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 91.5%

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

      1. neg-mul-1 91.5%

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

   6. Simplified 91.5%

      \[\leadsto x + \frac{0.3333333333333333}{z} \cdot \color{blue}{\left(-y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-8}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 0.0112:\\ \;\;\;\;x + \frac{t}{3 \cdot \left(y \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{0.3333333333333333}{z}\\ \end{array} \]

Alternative 6: 89.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-8}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 0.00016:\\ \;\;\;\;x + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{0.3333333333333333}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.2e-8)
   (- x (/ y (* z 3.0)))
   (if (<= y 0.00016)
     (+ x (/ t (* z (* y 3.0))))
     (- x (* y (/ 0.3333333333333333 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.2e-8) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 0.00016) {
		tmp = x + (t / (z * (y * 3.0)));
	} else {
		tmp = x - (y * (0.3333333333333333 / z));
	}
	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 <= (-5.2d-8)) then
        tmp = x - (y / (z * 3.0d0))
    else if (y <= 0.00016d0) then
        tmp = x + (t / (z * (y * 3.0d0)))
    else
        tmp = x - (y * (0.3333333333333333d0 / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.2e-8) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 0.00016) {
		tmp = x + (t / (z * (y * 3.0)));
	} else {
		tmp = x - (y * (0.3333333333333333 / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -5.2e-8:
		tmp = x - (y / (z * 3.0))
	elif y <= 0.00016:
		tmp = x + (t / (z * (y * 3.0)))
	else:
		tmp = x - (y * (0.3333333333333333 / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.2e-8)
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	elseif (y <= 0.00016)
		tmp = Float64(x + Float64(t / Float64(z * Float64(y * 3.0))));
	else
		tmp = Float64(x - Float64(y * Float64(0.3333333333333333 / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -5.2e-8)
		tmp = x - (y / (z * 3.0));
	elseif (y <= 0.00016)
		tmp = x + (t / (z * (y * 3.0)));
	else
		tmp = x - (y * (0.3333333333333333 / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5.2e-8], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00016], N[(x + N[(t / N[(z * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.2000000000000002e-8

   1. Initial program 98.4%

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

      1. associate-*l* 98.4%

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

      2. *-commutative 98.4%

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

   3. Simplified 98.4%

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

      1. associate-+l- 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-*l* 98.4%

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

      4. associate-/l/ 98.4%

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

      5. sub-div 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around inf 89.3%

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

   if -5.2000000000000002e-8 < y < 1.60000000000000013e-4

   1. Initial program 96.2%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in t around inf 94.8%

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

      1. associate-*r/ 94.9%

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

      2. *-commutative 94.9%

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

      3. frac-times 90.5%

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

      4. clear-num 90.4%

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

      5. frac-times 94.9%

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

      6. *-un-lft-identity 94.9%

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

      7. div-inv 95.0%

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

      8. metadata-eval 95.0%

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

      9. associate-*l* 94.9%

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

   6. Applied egg-rr 94.9%

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

   if 1.60000000000000013e-4 < y

   1. Initial program 97.2%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 91.5%

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

      1. neg-mul-1 91.5%

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

   6. Simplified 91.5%

      \[\leadsto x + \frac{0.3333333333333333}{z} \cdot \color{blue}{\left(-y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-8}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 0.00016:\\ \;\;\;\;x + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{0.3333333333333333}{z}\\ \end{array} \]

Alternative 7: 92.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{-7}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{-7}:\\ \;\;\;\;x + \frac{t \cdot \frac{0.3333333333333333}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{0.3333333333333333}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.18e-7)
   (- x (/ y (* z 3.0)))
   (if (<= y 6.9e-7)
     (+ x (/ (* t (/ 0.3333333333333333 z)) y))
     (- x (* y (/ 0.3333333333333333 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.18e-7) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 6.9e-7) {
		tmp = x + ((t * (0.3333333333333333 / z)) / y);
	} else {
		tmp = x - (y * (0.3333333333333333 / z));
	}
	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 <= (-1.18d-7)) then
        tmp = x - (y / (z * 3.0d0))
    else if (y <= 6.9d-7) then
        tmp = x + ((t * (0.3333333333333333d0 / z)) / y)
    else
        tmp = x - (y * (0.3333333333333333d0 / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.18e-7) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 6.9e-7) {
		tmp = x + ((t * (0.3333333333333333 / z)) / y);
	} else {
		tmp = x - (y * (0.3333333333333333 / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.18e-7:
		tmp = x - (y / (z * 3.0))
	elif y <= 6.9e-7:
		tmp = x + ((t * (0.3333333333333333 / z)) / y)
	else:
		tmp = x - (y * (0.3333333333333333 / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.18e-7)
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	elseif (y <= 6.9e-7)
		tmp = Float64(x + Float64(Float64(t * Float64(0.3333333333333333 / z)) / y));
	else
		tmp = Float64(x - Float64(y * Float64(0.3333333333333333 / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.18e-7)
		tmp = x - (y / (z * 3.0));
	elseif (y <= 6.9e-7)
		tmp = x + ((t * (0.3333333333333333 / z)) / y);
	else
		tmp = x - (y * (0.3333333333333333 / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.18e-7], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.9e-7], N[(x + N[(N[(t * N[(0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.18e-7

   1. Initial program 98.4%

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

      1. associate-*l* 98.4%

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

      2. *-commutative 98.4%

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

   3. Simplified 98.4%

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

      1. associate-+l- 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-*l* 98.4%

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

      4. associate-/l/ 98.4%

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

      5. sub-div 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around inf 89.3%

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

   if -1.18e-7 < y < 6.8999999999999996e-7

   1. Initial program 96.2%

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

      1. associate-*l* 96.1%

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

      2. *-commutative 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in y around 0 94.8%

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

      1. +-commutative 94.8%

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

      2. *-commutative 94.8%

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

      3. associate-/r* 90.5%

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

      4. associate-*l/ 90.5%

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

      5. associate-*r/ 90.5%

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

      6. associate-*l/ 95.3%

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

   6. Simplified 95.3%

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

   if 6.8999999999999996e-7 < y

   1. Initial program 97.2%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 91.5%

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

      1. neg-mul-1 91.5%

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

   6. Simplified 91.5%

      \[\leadsto x + \frac{0.3333333333333333}{z} \cdot \color{blue}{\left(-y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{-7}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{-7}:\\ \;\;\;\;x + \frac{t \cdot \frac{0.3333333333333333}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{0.3333333333333333}{z}\\ \end{array} \]

Alternative 8: 92.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 0.145:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{0.3333333333333333}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.2e-10)
   (- x (/ y (* z 3.0)))
   (if (<= y 0.145)
     (+ x (/ (/ t (* z 3.0)) y))
     (- x (* y (/ 0.3333333333333333 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.2e-10) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 0.145) {
		tmp = x + ((t / (z * 3.0)) / y);
	} else {
		tmp = x - (y * (0.3333333333333333 / z));
	}
	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 <= (-3.2d-10)) then
        tmp = x - (y / (z * 3.0d0))
    else if (y <= 0.145d0) then
        tmp = x + ((t / (z * 3.0d0)) / y)
    else
        tmp = x - (y * (0.3333333333333333d0 / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.2e-10) {
		tmp = x - (y / (z * 3.0));
	} else if (y <= 0.145) {
		tmp = x + ((t / (z * 3.0)) / y);
	} else {
		tmp = x - (y * (0.3333333333333333 / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -3.2e-10:
		tmp = x - (y / (z * 3.0))
	elif y <= 0.145:
		tmp = x + ((t / (z * 3.0)) / y)
	else:
		tmp = x - (y * (0.3333333333333333 / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.2e-10)
		tmp = Float64(x - Float64(y / Float64(z * 3.0)));
	elseif (y <= 0.145)
		tmp = Float64(x + Float64(Float64(t / Float64(z * 3.0)) / y));
	else
		tmp = Float64(x - Float64(y * Float64(0.3333333333333333 / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -3.2e-10)
		tmp = x - (y / (z * 3.0));
	elseif (y <= 0.145)
		tmp = x + ((t / (z * 3.0)) / y);
	else
		tmp = x - (y * (0.3333333333333333 / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.2e-10], N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.145], N[(x + N[(N[(t / N[(z * 3.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.19999999999999981e-10

   1. Initial program 98.4%

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

      1. associate-*l* 98.4%

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

      2. *-commutative 98.4%

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

   3. Simplified 98.4%

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

      1. associate-+l- 98.4%

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

      2. *-commutative 98.4%

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

      3. associate-*l* 98.4%

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

      4. associate-/l/ 98.4%

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

      5. sub-div 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around inf 89.3%

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

   if -3.19999999999999981e-10 < y < 0.14499999999999999

   1. Initial program 96.2%

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

      1. associate-*l* 96.1%

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

      2. *-commutative 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in y around 0 94.8%

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

      1. +-commutative 94.8%

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

      2. *-commutative 94.8%

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

      3. associate-/r* 90.5%

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

      4. associate-*l/ 90.5%

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

      5. associate-*r/ 90.5%

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

      6. associate-*l/ 95.3%

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

   6. Simplified 95.3%

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

   7. Taylor expanded in t around 0 95.4%

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

      1. associate-*r/ 95.3%

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

      2. *-commutative 95.3%

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

      3. /-rgt-identity 95.3%

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

      4. associate-/l* 95.4%

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

      5. metadata-eval 95.4%

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

      6. associate-/l/ 95.5%

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

   9. Simplified 95.5%

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

   if 0.14499999999999999 < y

   1. Initial program 97.2%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around 0 91.5%

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

      1. neg-mul-1 91.5%

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

   6. Simplified 91.5%

      \[\leadsto x + \frac{0.3333333333333333}{z} \cdot \color{blue}{\left(-y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 0.145:\\ \;\;\;\;x + \frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{0.3333333333333333}{z}\\ \end{array} \]

Alternative 9: 95.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.1%

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

3. Simplified 96.1%

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

4. Final simplification 96.1%

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

Alternative 10: 95.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x + (((t / y) - 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 + (((t / y) - y) / (z * 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.1%

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

   1. associate-*l* 97.1%

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

   2. *-commutative 97.1%

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

3. Simplified 97.1%

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

   1. associate-+l- 97.1%

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

   2. *-commutative 97.1%

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

   3. associate-*l* 97.1%

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

   4. associate-/l/ 95.3%

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

   5. sub-div 96.1%

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

5. Applied egg-rr 96.1%

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

6. Final simplification 96.1%

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

Alternative 11: 47.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.38 \lor \neg \left(y \leq 1.62 \cdot 10^{-44}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -0.38) (not (<= y 1.62e-44)))
   (* y (/ -0.3333333333333333 z))
   x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -0.38) || !(y <= 1.62e-44)) {
		tmp = y * (-0.3333333333333333 / z);
	} else {
		tmp = x;
	}
	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 <= (-0.38d0)) .or. (.not. (y <= 1.62d-44))) then
        tmp = y * ((-0.3333333333333333d0) / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -0.38) || !(y <= 1.62e-44)) {
		tmp = y * (-0.3333333333333333 / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -0.38) or not (y <= 1.62e-44):
		tmp = y * (-0.3333333333333333 / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -0.38) || !(y <= 1.62e-44))
		tmp = Float64(y * Float64(-0.3333333333333333 / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -0.38) || ~((y <= 1.62e-44)))
		tmp = y * (-0.3333333333333333 / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -0.38], N[Not[LessEqual[y, 1.62e-44]], $MachinePrecision]], N[(y * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -0.38 or 1.6200000000000001e-44 < y

   1. Initial program 97.9%

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

      1. clear-num 97.9%

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

      2. inv-pow 97.9%

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

      3. *-commutative 97.9%

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

      4. *-un-lft-identity 97.9%

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

      5. times-frac 97.8%

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

      6. metadata-eval 97.8%

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

   3. Applied egg-rr 97.8%

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

   4. Taylor expanded in y around inf 68.8%

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

      1. *-commutative 68.8%

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

   6. Simplified 68.8%

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

   7. Taylor expanded in y around 0 68.8%

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

      1. associate-*r/ 68.8%

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

      2. associate-*l/ 68.8%

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

      3. metadata-eval 68.8%

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

      4. distribute-neg-frac 68.8%

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

      5. *-commutative 68.8%

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

      6. distribute-neg-frac 68.8%

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

      7. metadata-eval 68.8%

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

   9. Simplified 68.8%

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

   if -0.38 < y < 1.6200000000000001e-44

   1. Initial program 96.2%

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

      1. associate-*l* 96.2%

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

      2. *-commutative 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in x around inf 30.8%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.38 \lor \neg \left(y \leq 1.62 \cdot 10^{-44}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12: 47.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{-44}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -6.5)
   (/ -0.3333333333333333 (/ z y))
   (if (<= y 1.62e-44) x (* y (/ -0.3333333333333333 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.5) {
		tmp = -0.3333333333333333 / (z / y);
	} else if (y <= 1.62e-44) {
		tmp = x;
	} else {
		tmp = y * (-0.3333333333333333 / z);
	}
	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 <= (-6.5d0)) then
        tmp = (-0.3333333333333333d0) / (z / y)
    else if (y <= 1.62d-44) then
        tmp = x
    else
        tmp = y * ((-0.3333333333333333d0) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.5) {
		tmp = -0.3333333333333333 / (z / y);
	} else if (y <= 1.62e-44) {
		tmp = x;
	} else {
		tmp = y * (-0.3333333333333333 / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -6.5:
		tmp = -0.3333333333333333 / (z / y)
	elif y <= 1.62e-44:
		tmp = x
	else:
		tmp = y * (-0.3333333333333333 / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -6.5)
		tmp = Float64(-0.3333333333333333 / Float64(z / y));
	elseif (y <= 1.62e-44)
		tmp = x;
	else
		tmp = Float64(y * Float64(-0.3333333333333333 / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -6.5)
		tmp = -0.3333333333333333 / (z / y);
	elseif (y <= 1.62e-44)
		tmp = x;
	else
		tmp = y * (-0.3333333333333333 / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -6.5], N[(-0.3333333333333333 / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.62e-44], x, N[(y * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.5

   1. Initial program 98.4%

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

      1. clear-num 98.3%

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

      2. inv-pow 98.3%

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

      3. *-commutative 98.3%

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

      4. *-un-lft-identity 98.3%

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

      5. times-frac 98.3%

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

      6. metadata-eval 98.3%

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

   3. Applied egg-rr 98.3%

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

   4. Taylor expanded in y around inf 70.8%

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

      1. *-commutative 70.8%

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

   6. Simplified 70.8%

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

      1. *-commutative 70.8%

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

      2. clear-num 70.8%

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

      3. un-div-inv 70.8%

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

   8. Applied egg-rr 70.8%

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

   if -6.5 < y < 1.6200000000000001e-44

   1. Initial program 96.2%

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

      1. associate-*l* 96.2%

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

      2. *-commutative 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in x around inf 30.8%

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

   if 1.6200000000000001e-44 < y

   1. Initial program 97.3%

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

      1. clear-num 97.3%

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

      2. inv-pow 97.3%

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

      3. *-commutative 97.3%

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

      4. *-un-lft-identity 97.3%

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

      5. times-frac 97.3%

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

      6. metadata-eval 97.3%

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

   3. Applied egg-rr 97.3%

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

   4. Taylor expanded in y around inf 66.6%

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

      1. *-commutative 66.6%

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

   6. Simplified 66.6%

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

   7. Taylor expanded in y around 0 66.6%

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

      1. associate-*r/ 66.6%

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

      2. associate-*l/ 66.7%

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

      3. metadata-eval 66.7%

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

      4. distribute-neg-frac 66.7%

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

      5. *-commutative 66.7%

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

      6. distribute-neg-frac 66.7%

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

      7. metadata-eval 66.7%

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

   9. Simplified 66.7%

      \[\leadsto \color{blue}{y \cdot \frac{-0.3333333333333333}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{z}{y}}\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{-44}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 13: 47.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0105:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{-44}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -0.0105)
   (/ y (* z -3.0))
   (if (<= y 1.62e-44) x (* y (/ -0.3333333333333333 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -0.0105) {
		tmp = y / (z * -3.0);
	} else if (y <= 1.62e-44) {
		tmp = x;
	} else {
		tmp = y * (-0.3333333333333333 / z);
	}
	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 <= (-0.0105d0)) then
        tmp = y / (z * (-3.0d0))
    else if (y <= 1.62d-44) then
        tmp = x
    else
        tmp = y * ((-0.3333333333333333d0) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -0.0105) {
		tmp = y / (z * -3.0);
	} else if (y <= 1.62e-44) {
		tmp = x;
	} else {
		tmp = y * (-0.3333333333333333 / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -0.0105:
		tmp = y / (z * -3.0)
	elif y <= 1.62e-44:
		tmp = x
	else:
		tmp = y * (-0.3333333333333333 / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -0.0105)
		tmp = Float64(y / Float64(z * -3.0));
	elseif (y <= 1.62e-44)
		tmp = x;
	else
		tmp = Float64(y * Float64(-0.3333333333333333 / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -0.0105)
		tmp = y / (z * -3.0);
	elseif (y <= 1.62e-44)
		tmp = x;
	else
		tmp = y * (-0.3333333333333333 / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -0.0105], N[(y / N[(z * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.62e-44], x, N[(y * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0105000000000000007

   1. Initial program 98.4%

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

      1. clear-num 98.3%

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

      2. inv-pow 98.3%

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

      3. *-commutative 98.3%

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

      4. *-un-lft-identity 98.3%

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

      5. times-frac 98.3%

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

      6. metadata-eval 98.3%

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

   3. Applied egg-rr 98.3%

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

   4. Taylor expanded in y around inf 70.8%

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

      1. *-commutative 70.8%

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

   6. Simplified 70.8%

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

      1. frac-2neg 70.8%

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

      2. associate-*l/ 70.7%

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

      3. associate-*r/ 70.8%

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

      4. clear-num 70.7%

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

      5. metadata-eval 70.7%

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

      6. frac-2neg 70.7%

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

      7. div-inv 70.7%

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

      8. metadata-eval 70.7%

         \[\leadsto \left(-y\right) \cdot \frac{1}{z \cdot \color{blue}{3}} \]

      9. un-div-inv 70.9%

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

      10. add-sqr-sqrt 70.7%

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

      11. sqrt-unprod 46.2%

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

      12. sqr-neg 46.2%

          \[\leadsto \frac{\sqrt{\color{blue}{y \cdot y}}}{z \cdot 3} \]

      13. sqrt-unprod 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{z \cdot 3} \]

      14. add-sqr-sqrt 2.8%

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

      15. frac-2neg 2.8%

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

      16. add-sqr-sqrt 2.8%

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

      17. sqrt-unprod 2.4%

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

      18. sqr-neg 2.4%

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

      19. sqrt-unprod 0.0%

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

      20. add-sqr-sqrt 70.9%

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

      21. distribute-rgt-neg-in 70.9%

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

      22. metadata-eval 70.9%

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

   8. Applied egg-rr 70.9%

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

   if -0.0105000000000000007 < y < 1.6200000000000001e-44

   1. Initial program 96.2%

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

      1. associate-*l* 96.2%

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

      2. *-commutative 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in x around inf 30.8%

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

   if 1.6200000000000001e-44 < y

   1. Initial program 97.3%

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

      1. clear-num 97.3%

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

      2. inv-pow 97.3%

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

      3. *-commutative 97.3%

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

      4. *-un-lft-identity 97.3%

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

      5. times-frac 97.3%

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

      6. metadata-eval 97.3%

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

   3. Applied egg-rr 97.3%

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

   4. Taylor expanded in y around inf 66.6%

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

      1. *-commutative 66.6%

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

   6. Simplified 66.6%

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

   7. Taylor expanded in y around 0 66.6%

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

      1. associate-*r/ 66.6%

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

      2. associate-*l/ 66.7%

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

      3. metadata-eval 66.7%

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

      4. distribute-neg-frac 66.7%

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

      5. *-commutative 66.7%

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

      6. distribute-neg-frac 66.7%

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

      7. metadata-eval 66.7%

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

   9. Simplified 66.7%

      \[\leadsto \color{blue}{y \cdot \frac{-0.3333333333333333}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0105:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{-44}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 14: 64.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x - (0.3333333333333333 * (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 / z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.1%

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

   1. associate-*l* 97.1%

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

   2. *-commutative 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in t around 0 63.6%

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

5. Final simplification 63.6%

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

Alternative 15: 64.5% 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 97.1%

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

   1. associate-*l* 97.1%

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

   2. *-commutative 97.1%

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

3. Simplified 97.1%

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

   1. associate-+l- 97.1%

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

   2. *-commutative 97.1%

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

   3. associate-*l* 97.1%

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

   4. associate-/l/ 95.3%

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

   5. sub-div 96.1%

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

5. Applied egg-rr 96.1%

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

6. Taylor expanded in y around inf 63.6%

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

7. Final simplification 63.6%

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

Alternative 16: 64.4% 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 97.1%

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

3. Simplified 96.1%

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

4. Taylor expanded in t around 0 63.6%

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

   1. neg-mul-1 63.6%

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

6. Simplified 63.6%

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

7. Final simplification 63.6%

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

Alternative 17: 30.2% 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 97.1%

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

   1. associate-*l* 97.1%

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

   2. *-commutative 97.1%

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

3. Simplified 97.1%

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

4. Taylor expanded in x around inf 26.0%

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

5. Final simplification 26.0%

   \[\leadsto x \]

Developer target: 96.1% 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 2023340 
(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))))