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

Percentage Accurate: 95.7% → 97.7%

Time: 9.9s

Alternatives: 13

Speedup: 1.4×

• 27.1% 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: 95.7% 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: 97.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 2 \cdot 10^{+290}:\\ \;\;\;\;x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{z \cdot \left(y \cdot 3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{0.3333333333333333}{z} \cdot \left(\frac{t}{y} - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (- x (/ y (* z 3.0))) (/ t (* y (* z 3.0)))) 2e+290)
   (+ x (fma -0.3333333333333333 (/ y z) (/ t (* z (* y 3.0)))))
   (+ x (* (/ 0.3333333333333333 z) (- (/ t y) y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - (y / (z * 3.0))) + (t / (y * (z * 3.0)))) <= 2e+290) {
		tmp = x + fma(-0.3333333333333333, (y / z), (t / (z * (y * 3.0))));
	} else {
		tmp = x + ((0.3333333333333333 / z) * ((t / y) - y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(y * Float64(z * 3.0)))) <= 2e+290)
		tmp = Float64(x + fma(-0.3333333333333333, Float64(y / z), Float64(t / Float64(z * Float64(y * 3.0)))));
	else
		tmp = Float64(x + Float64(Float64(0.3333333333333333 / z) * Float64(Float64(t / y) - y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+290], N[(x + N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + N[(t / N[(z * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(0.3333333333333333 / z), $MachinePrecision] * N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y))) < 2.00000000000000012e290

   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. sub-neg 97.9%

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

      2. associate-+l+ 97.9%

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

      3. distribute-frac-neg 97.9%

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

      4. neg-mul-1 97.9%

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

      5. *-commutative 97.9%

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

      6. times-frac 97.9%

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

      7. fma-define 97.9%

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

      8. metadata-eval 97.9%

         \[\leadsto x + \mathsf{fma}\left(\color{blue}{-0.3333333333333333}, \frac{y}{z}, \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      9. associate-*l* 97.9%

         \[\leadsto x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{\color{blue}{z \cdot \left(3 \cdot y\right)}}\right) \]

      10. *-commutative 97.9%

          \[\leadsto x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{z \cdot \color{blue}{\left(y \cdot 3\right)}}\right) \]

   3. Simplified 97.9%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{z \cdot \left(y \cdot 3\right)}\right)} \]
   4. Add Preprocessing

   if 2.00000000000000012e290 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 90.8%

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

      1. +-commutative 90.8%

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

      2. associate-+r- 90.8%

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

      3. +-commutative 90.8%

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

      4. associate--l+ 90.8%

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

      5. sub-neg 90.8%

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

      6. remove-double-neg 90.8%

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

      7. distribute-frac-neg 90.8%

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

      8. distribute-neg-in 90.8%

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

      9. remove-double-neg 90.8%

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

      10. sub-neg 90.8%

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

      11. neg-mul-1 90.8%

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

      12. times-frac 96.2%

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

      13. distribute-frac-neg 96.2%

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

      14. neg-mul-1 96.2%

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

      15. *-commutative 96.2%

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

      16. associate-/l* 96.2%

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

      17. *-commutative 96.2%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{0.3333333333333333}{z} \cdot \left(\frac{t}{y} - y\right)} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 2 \cdot 10^{+290}:\\ \;\;\;\;x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{z \cdot \left(y \cdot 3\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{0.3333333333333333}{z} \cdot \left(\frac{t}{y} - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{-0.3333333333333333}{z}\\ t_2 := t \cdot \frac{0.3333333333333333}{y \cdot z}\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+39}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1820000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-158}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-116}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+14}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (/ -0.3333333333333333 z)))
        (t_2 (* t (/ 0.3333333333333333 (* y z)))))
   (if (<= y -1.7e+101)
     t_1
     (if (<= y -3e+39)
       x
       (if (<= y -1820000000000.0)
         t_1
         (if (<= y -4e-158)
           x
           (if (<= y 2.5e-116)
             t_2
             (if (<= y 4.5e-93)
               x
               (if (<= y 1.75e+14) t_2 (* -0.3333333333333333 (/ y z)))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (-0.3333333333333333 / z);
	double t_2 = t * (0.3333333333333333 / (y * z));
	double tmp;
	if (y <= -1.7e+101) {
		tmp = t_1;
	} else if (y <= -3e+39) {
		tmp = x;
	} else if (y <= -1820000000000.0) {
		tmp = t_1;
	} else if (y <= -4e-158) {
		tmp = x;
	} else if (y <= 2.5e-116) {
		tmp = t_2;
	} else if (y <= 4.5e-93) {
		tmp = x;
	} else if (y <= 1.75e+14) {
		tmp = t_2;
	} else {
		tmp = -0.3333333333333333 * (y / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = y * ((-0.3333333333333333d0) / z)
    t_2 = t * (0.3333333333333333d0 / (y * z))
    if (y <= (-1.7d+101)) then
        tmp = t_1
    else if (y <= (-3d+39)) then
        tmp = x
    else if (y <= (-1820000000000.0d0)) then
        tmp = t_1
    else if (y <= (-4d-158)) then
        tmp = x
    else if (y <= 2.5d-116) then
        tmp = t_2
    else if (y <= 4.5d-93) then
        tmp = x
    else if (y <= 1.75d+14) then
        tmp = t_2
    else
        tmp = (-0.3333333333333333d0) * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * (-0.3333333333333333 / z);
	double t_2 = t * (0.3333333333333333 / (y * z));
	double tmp;
	if (y <= -1.7e+101) {
		tmp = t_1;
	} else if (y <= -3e+39) {
		tmp = x;
	} else if (y <= -1820000000000.0) {
		tmp = t_1;
	} else if (y <= -4e-158) {
		tmp = x;
	} else if (y <= 2.5e-116) {
		tmp = t_2;
	} else if (y <= 4.5e-93) {
		tmp = x;
	} else if (y <= 1.75e+14) {
		tmp = t_2;
	} else {
		tmp = -0.3333333333333333 * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (-0.3333333333333333 / z)
	t_2 = t * (0.3333333333333333 / (y * z))
	tmp = 0
	if y <= -1.7e+101:
		tmp = t_1
	elif y <= -3e+39:
		tmp = x
	elif y <= -1820000000000.0:
		tmp = t_1
	elif y <= -4e-158:
		tmp = x
	elif y <= 2.5e-116:
		tmp = t_2
	elif y <= 4.5e-93:
		tmp = x
	elif y <= 1.75e+14:
		tmp = t_2
	else:
		tmp = -0.3333333333333333 * (y / z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(-0.3333333333333333 / z))
	t_2 = Float64(t * Float64(0.3333333333333333 / Float64(y * z)))
	tmp = 0.0
	if (y <= -1.7e+101)
		tmp = t_1;
	elseif (y <= -3e+39)
		tmp = x;
	elseif (y <= -1820000000000.0)
		tmp = t_1;
	elseif (y <= -4e-158)
		tmp = x;
	elseif (y <= 2.5e-116)
		tmp = t_2;
	elseif (y <= 4.5e-93)
		tmp = x;
	elseif (y <= 1.75e+14)
		tmp = t_2;
	else
		tmp = Float64(-0.3333333333333333 * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (-0.3333333333333333 / z);
	t_2 = t * (0.3333333333333333 / (y * z));
	tmp = 0.0;
	if (y <= -1.7e+101)
		tmp = t_1;
	elseif (y <= -3e+39)
		tmp = x;
	elseif (y <= -1820000000000.0)
		tmp = t_1;
	elseif (y <= -4e-158)
		tmp = x;
	elseif (y <= 2.5e-116)
		tmp = t_2;
	elseif (y <= 4.5e-93)
		tmp = x;
	elseif (y <= 1.75e+14)
		tmp = t_2;
	else
		tmp = -0.3333333333333333 * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(-0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(0.3333333333333333 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.7e+101], t$95$1, If[LessEqual[y, -3e+39], x, If[LessEqual[y, -1820000000000.0], t$95$1, If[LessEqual[y, -4e-158], x, If[LessEqual[y, 2.5e-116], t$95$2, If[LessEqual[y, 4.5e-93], x, If[LessEqual[y, 1.75e+14], t$95$2, N[(-0.3333333333333333 * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.70000000000000009e101 or -3e39 < y < -1.82e12

   1. Initial program 95.5%

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

      1. +-commutative 95.5%

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

      2. associate-+r- 95.5%

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

      3. +-commutative 95.5%

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

      4. associate--l+ 95.5%

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

      5. sub-neg 95.5%

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

      6. remove-double-neg 95.5%

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

      7. distribute-frac-neg 95.5%

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

      8. distribute-neg-in 95.5%

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

      9. remove-double-neg 95.5%

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

      10. sub-neg 95.5%

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

      11. neg-mul-1 95.5%

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

      12. times-frac 95.5%

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

      13. distribute-frac-neg 95.5%

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

      14. neg-mul-1 95.5%

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

      15. *-commutative 95.5%

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

      16. associate-/l* 95.4%

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

      17. *-commutative 95.4%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 98.6%

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

      1. associate-*r/ 98.7%

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

      2. metadata-eval 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in x around 0 78.6%

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

   if -1.70000000000000009e101 < y < -3e39 or -1.82e12 < y < -4.00000000000000026e-158 or 2.5000000000000001e-116 < y < 4.5000000000000002e-93

   1. Initial program 98.0%

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

      1. +-commutative 98.0%

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

      2. associate-+r- 98.0%

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

      3. +-commutative 98.0%

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

      4. associate--l+ 98.0%

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

      5. sub-neg 98.0%

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

      6. remove-double-neg 98.0%

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

      7. distribute-frac-neg 98.0%

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

      8. distribute-neg-in 98.0%

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

      9. remove-double-neg 98.0%

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

      10. sub-neg 98.0%

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

      11. neg-mul-1 98.0%

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

      12. times-frac 92.8%

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

      13. distribute-frac-neg 92.8%

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

      14. neg-mul-1 92.8%

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

      15. *-commutative 92.8%

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

      16. associate-/l* 92.8%

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

      17. *-commutative 92.8%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in x around inf 60.2%

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

   if -4.00000000000000026e-158 < y < 2.5000000000000001e-116 or 4.5000000000000002e-93 < y < 1.75e14

   1. Initial program 94.2%

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

      1. +-commutative 94.2%

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

      2. associate-+r- 94.2%

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

      3. +-commutative 94.2%

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

      4. associate--l+ 94.2%

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

      5. sub-neg 94.2%

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

      6. remove-double-neg 94.2%

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

      7. distribute-frac-neg 94.2%

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

      8. distribute-neg-in 94.2%

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

      9. remove-double-neg 94.2%

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

      10. sub-neg 94.2%

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

      11. neg-mul-1 94.2%

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

      12. times-frac 91.4%

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

      13. distribute-frac-neg 91.4%

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

      14. neg-mul-1 91.4%

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

      15. *-commutative 91.4%

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

      16. associate-/l* 91.4%

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

      17. *-commutative 91.4%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in y around 0 94.7%

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

   6. Taylor expanded in t around inf 75.1%

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

      1. associate-*r/ 75.1%

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

      2. associate-*l/ 75.1%

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

      3. *-commutative 75.1%

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

   8. Simplified 75.1%

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

      1. associate-/l* 69.6%

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

      2. *-commutative 69.6%

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

      3. associate-/l/ 69.6%

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

   10. Applied egg-rr 69.6%

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

   if 1.75e14 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0 95.2%

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

   4. Taylor expanded in x around 0 66.8%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{y}{z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+101}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq -3 \cdot 10^{+39}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1820000000000:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-158}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-116}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{y \cdot z}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+14}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\left(x + \frac{t}{z \cdot \left(y \cdot 3\right)}\right) + \frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{0.3333333333333333}{z} \cdot \left(\frac{t}{y} - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (- x (/ y (* z 3.0))) (/ t (* y (* z 3.0)))) 2e+290)
   (+ (+ x (/ t (* z (* y 3.0)))) (/ y (* z -3.0)))
   (+ x (* (/ 0.3333333333333333 z) (- (/ t y) y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - (y / (z * 3.0))) + (t / (y * (z * 3.0)))) <= 2e+290) {
		tmp = (x + (t / (z * (y * 3.0)))) + (y / (z * -3.0));
	} else {
		tmp = x + ((0.3333333333333333 / z) * ((t / y) - 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 (((x - (y / (z * 3.0d0))) + (t / (y * (z * 3.0d0)))) <= 2d+290) then
        tmp = (x + (t / (z * (y * 3.0d0)))) + (y / (z * (-3.0d0)))
    else
        tmp = x + ((0.3333333333333333d0 / z) * ((t / y) - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - (y / (z * 3.0))) + (t / (y * (z * 3.0)))) <= 2e+290) {
		tmp = (x + (t / (z * (y * 3.0)))) + (y / (z * -3.0));
	} else {
		tmp = x + ((0.3333333333333333 / z) * ((t / y) - y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x - (y / (z * 3.0))) + (t / (y * (z * 3.0)))) <= 2e+290:
		tmp = (x + (t / (z * (y * 3.0)))) + (y / (z * -3.0))
	else:
		tmp = x + ((0.3333333333333333 / z) * ((t / y) - y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(y * Float64(z * 3.0)))) <= 2e+290)
		tmp = Float64(Float64(x + Float64(t / Float64(z * Float64(y * 3.0)))) + Float64(y / Float64(z * -3.0)));
	else
		tmp = Float64(x + Float64(Float64(0.3333333333333333 / z) * Float64(Float64(t / y) - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x - (y / (z * 3.0))) + (t / (y * (z * 3.0)))) <= 2e+290)
		tmp = (x + (t / (z * (y * 3.0)))) + (y / (z * -3.0));
	else
		tmp = x + ((0.3333333333333333 / z) * ((t / y) - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+290], N[(N[(x + N[(t / N[(z * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y / N[(z * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(0.3333333333333333 / z), $MachinePrecision] * N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y))) < 2.00000000000000012e290

   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. +-commutative 97.9%

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

      2. associate-+r- 97.9%

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

      3. sub-neg 97.9%

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

      4. associate-*l* 97.9%

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

      5. *-commutative 97.9%

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

      6. distribute-frac-neg2 97.9%

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

      7. distribute-rgt-neg-in 97.9%

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

      8. metadata-eval 97.9%

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

   3. Simplified 97.9%

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

   if 2.00000000000000012e290 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 90.8%

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

      1. +-commutative 90.8%

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

      2. associate-+r- 90.8%

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

      3. +-commutative 90.8%

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

      4. associate--l+ 90.8%

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

      5. sub-neg 90.8%

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

      6. remove-double-neg 90.8%

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

      7. distribute-frac-neg 90.8%

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

      8. distribute-neg-in 90.8%

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

      9. remove-double-neg 90.8%

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

      10. sub-neg 90.8%

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

      11. neg-mul-1 90.8%

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

      12. times-frac 96.2%

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

      13. distribute-frac-neg 96.2%

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

      14. neg-mul-1 96.2%

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

      15. *-commutative 96.2%

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

      16. associate-/l* 96.2%

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

      17. *-commutative 96.2%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x + \frac{0.3333333333333333}{z} \cdot \left(\frac{t}{y} - y\right)} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\left(x + \frac{t}{z \cdot \left(y \cdot 3\right)}\right) + \frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{0.3333333333333333}{z} \cdot \left(\frac{t}{y} - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-82} \lor \neg \left(y \leq 2.46 \cdot 10^{-88}\right):\\ \;\;\;\;x + \frac{0.3333333333333333}{z} \cdot \left(\frac{t}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \frac{t}{z} + x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1e-82) (not (<= y 2.46e-88)))
   (+ x (* (/ 0.3333333333333333 z) (- (/ t y) y)))
   (/ (+ (* 0.3333333333333333 (/ t z)) (* x y)) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1e-82) || !(y <= 2.46e-88)) {
		tmp = x + ((0.3333333333333333 / z) * ((t / y) - y));
	} else {
		tmp = ((0.3333333333333333 * (t / z)) + (x * y)) / 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 ((y <= (-1d-82)) .or. (.not. (y <= 2.46d-88))) then
        tmp = x + ((0.3333333333333333d0 / z) * ((t / y) - y))
    else
        tmp = ((0.3333333333333333d0 * (t / z)) + (x * y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1e-82) || !(y <= 2.46e-88)) {
		tmp = x + ((0.3333333333333333 / z) * ((t / y) - y));
	} else {
		tmp = ((0.3333333333333333 * (t / z)) + (x * y)) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1e-82) or not (y <= 2.46e-88):
		tmp = x + ((0.3333333333333333 / z) * ((t / y) - y))
	else:
		tmp = ((0.3333333333333333 * (t / z)) + (x * y)) / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1e-82) || !(y <= 2.46e-88))
		tmp = Float64(x + Float64(Float64(0.3333333333333333 / z) * Float64(Float64(t / y) - y)));
	else
		tmp = Float64(Float64(Float64(0.3333333333333333 * Float64(t / z)) + Float64(x * y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1e-82) || ~((y <= 2.46e-88)))
		tmp = x + ((0.3333333333333333 / z) * ((t / y) - y));
	else
		tmp = ((0.3333333333333333 * (t / z)) + (x * y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1e-82], N[Not[LessEqual[y, 2.46e-88]], $MachinePrecision]], N[(x + N[(N[(0.3333333333333333 / z), $MachinePrecision] * N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.3333333333333333 * N[(t / z), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1e-82 or 2.4599999999999999e-88 < y

   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. +-commutative 98.4%

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

      2. associate-+r- 98.4%

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

      3. +-commutative 98.4%

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

      4. associate--l+ 98.4%

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

      5. sub-neg 98.4%

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

      6. remove-double-neg 98.4%

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

      7. distribute-frac-neg 98.4%

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

      8. distribute-neg-in 98.4%

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

      9. remove-double-neg 98.4%

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

      10. sub-neg 98.4%

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

      11. neg-mul-1 98.4%

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

      12. times-frac 97.8%

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

      13. distribute-frac-neg 97.8%

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

      14. neg-mul-1 97.8%

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

      15. *-commutative 97.8%

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

      16. associate-/l* 97.9%

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

      17. *-commutative 97.9%

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

   3. Simplified 99.2%

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

   if -1e-82 < y < 2.4599999999999999e-88

   1. Initial program 93.2%

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

      1. +-commutative 93.2%

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

      2. associate-+r- 93.2%

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

      3. +-commutative 93.2%

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

      4. associate--l+ 93.2%

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

      5. sub-neg 93.2%

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

      6. remove-double-neg 93.2%

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

      7. distribute-frac-neg 93.2%

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

      8. distribute-neg-in 93.2%

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

      9. remove-double-neg 93.2%

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

      10. sub-neg 93.2%

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

      11. neg-mul-1 93.2%

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

      12. times-frac 88.4%

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

      13. distribute-frac-neg 88.4%

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

      14. neg-mul-1 88.4%

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

      15. *-commutative 88.4%

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

      16. associate-/l* 88.4%

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

      17. *-commutative 88.4%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in y around 0 96.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \frac{t}{z} + x \cdot y}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-82} \lor \neg \left(y \leq 2.46 \cdot 10^{-88}\right):\\ \;\;\;\;x + \frac{0.3333333333333333}{z} \cdot \left(\frac{t}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \frac{t}{z} + x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2200000000000:\\ \;\;\;\;x + \frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 23500000000:\\ \;\;\;\;x + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2200000000000.0)
   (+ x (/ (* y -0.3333333333333333) z))
   (if (<= y 23500000000.0)
     (+ x (/ t (* z (* y 3.0))))
     (- x (* (/ y z) 0.3333333333333333)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2200000000000.0) {
		tmp = x + ((y * -0.3333333333333333) / z);
	} else if (y <= 23500000000.0) {
		tmp = x + (t / (z * (y * 3.0)));
	} else {
		tmp = x - ((y / z) * 0.3333333333333333);
	}
	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 <= (-2200000000000.0d0)) then
        tmp = x + ((y * (-0.3333333333333333d0)) / z)
    else if (y <= 23500000000.0d0) then
        tmp = x + (t / (z * (y * 3.0d0)))
    else
        tmp = x - ((y / z) * 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2200000000000.0) {
		tmp = x + ((y * -0.3333333333333333) / z);
	} else if (y <= 23500000000.0) {
		tmp = x + (t / (z * (y * 3.0)));
	} else {
		tmp = x - ((y / z) * 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -2200000000000.0:
		tmp = x + ((y * -0.3333333333333333) / z)
	elif y <= 23500000000.0:
		tmp = x + (t / (z * (y * 3.0)))
	else:
		tmp = x - ((y / z) * 0.3333333333333333)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2200000000000.0)
		tmp = Float64(x + Float64(Float64(y * -0.3333333333333333) / z));
	elseif (y <= 23500000000.0)
		tmp = Float64(x + Float64(t / Float64(z * Float64(y * 3.0))));
	else
		tmp = Float64(x - Float64(Float64(y / z) * 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -2200000000000.0)
		tmp = x + ((y * -0.3333333333333333) / z);
	elseif (y <= 23500000000.0)
		tmp = x + (t / (z * (y * 3.0)));
	else
		tmp = x - ((y / z) * 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2200000000000.0], N[(x + N[(N[(y * -0.3333333333333333), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 23500000000.0], N[(x + N[(t / N[(z * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y / z), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2e12

   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. sub-neg 96.3%

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

      2. associate-+l+ 96.3%

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

      3. distribute-frac-neg 96.3%

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

      4. neg-mul-1 96.3%

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

      5. *-commutative 96.3%

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

      6. times-frac 96.3%

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

      7. fma-define 96.3%

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

      8. metadata-eval 96.3%

         \[\leadsto x + \mathsf{fma}\left(\color{blue}{-0.3333333333333333}, \frac{y}{z}, \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      9. associate-*l* 96.3%

         \[\leadsto x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{\color{blue}{z \cdot \left(3 \cdot y\right)}}\right) \]

      10. *-commutative 96.3%

          \[\leadsto x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{z \cdot \color{blue}{\left(y \cdot 3\right)}}\right) \]

   3. Simplified 96.3%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{z \cdot \left(y \cdot 3\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.8%

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

   6. Taylor expanded in y around inf 95.7%

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

      1. associate-*r/ 95.7%

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

      2. *-commutative 95.7%

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

   8. Simplified 95.7%

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

   if -2.2e12 < y < 2.35e10

   1. Initial program 95.2%

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

      1. +-commutative 95.2%

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

      2. associate-+r- 95.2%

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

      3. +-commutative 95.2%

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

      4. associate--l+ 95.2%

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

      5. sub-neg 95.2%

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

      6. remove-double-neg 95.2%

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

      7. distribute-frac-neg 95.2%

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

      8. distribute-neg-in 95.2%

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

      9. remove-double-neg 95.2%

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

      10. sub-neg 95.2%

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

      11. neg-mul-1 95.2%

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

      12. times-frac 91.2%

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

      13. distribute-frac-neg 91.2%

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

      14. neg-mul-1 91.2%

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

      15. *-commutative 91.2%

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

      16. associate-/l* 91.2%

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

      17. *-commutative 91.2%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in t around inf 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-/r* 85.7%

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

      3. associate-*l/ 85.7%

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

      4. metadata-eval 85.7%

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

      5. div-inv 85.7%

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

      6. div-inv 85.7%

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

      7. associate-/l/ 85.7%

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

      8. frac-times 89.7%

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

      9. *-commutative 89.7%

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

      10. *-un-lft-identity 89.7%

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

      11. *-commutative 89.7%

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

   7. Applied egg-rr 89.7%

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

   if 2.35e10 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0 95.2%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2200000000000:\\ \;\;\;\;x + \frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 23500000000:\\ \;\;\;\;x + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1950000000000:\\ \;\;\;\;x + \frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+14}:\\ \;\;\;\;x + 0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1950000000000.0)
   (+ x (/ (* y -0.3333333333333333) z))
   (if (<= y 6.5e+14)
     (+ x (* 0.3333333333333333 (/ t (* y z))))
     (- x (* (/ y z) 0.3333333333333333)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1950000000000.0) {
		tmp = x + ((y * -0.3333333333333333) / z);
	} else if (y <= 6.5e+14) {
		tmp = x + (0.3333333333333333 * (t / (y * z)));
	} else {
		tmp = x - ((y / z) * 0.3333333333333333);
	}
	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 <= (-1950000000000.0d0)) then
        tmp = x + ((y * (-0.3333333333333333d0)) / z)
    else if (y <= 6.5d+14) then
        tmp = x + (0.3333333333333333d0 * (t / (y * z)))
    else
        tmp = x - ((y / z) * 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1950000000000.0) {
		tmp = x + ((y * -0.3333333333333333) / z);
	} else if (y <= 6.5e+14) {
		tmp = x + (0.3333333333333333 * (t / (y * z)));
	} else {
		tmp = x - ((y / z) * 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1950000000000.0:
		tmp = x + ((y * -0.3333333333333333) / z)
	elif y <= 6.5e+14:
		tmp = x + (0.3333333333333333 * (t / (y * z)))
	else:
		tmp = x - ((y / z) * 0.3333333333333333)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1950000000000.0)
		tmp = Float64(x + Float64(Float64(y * -0.3333333333333333) / z));
	elseif (y <= 6.5e+14)
		tmp = Float64(x + Float64(0.3333333333333333 * Float64(t / Float64(y * z))));
	else
		tmp = Float64(x - Float64(Float64(y / z) * 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1950000000000.0)
		tmp = x + ((y * -0.3333333333333333) / z);
	elseif (y <= 6.5e+14)
		tmp = x + (0.3333333333333333 * (t / (y * z)));
	else
		tmp = x - ((y / z) * 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1950000000000.0], N[(x + N[(N[(y * -0.3333333333333333), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e+14], N[(x + N[(0.3333333333333333 * N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y / z), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.95e12

   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. sub-neg 96.3%

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

      2. associate-+l+ 96.3%

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

      3. distribute-frac-neg 96.3%

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

      4. neg-mul-1 96.3%

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

      5. *-commutative 96.3%

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

      6. times-frac 96.3%

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

      7. fma-define 96.3%

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

      8. metadata-eval 96.3%

         \[\leadsto x + \mathsf{fma}\left(\color{blue}{-0.3333333333333333}, \frac{y}{z}, \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      9. associate-*l* 96.3%

         \[\leadsto x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{\color{blue}{z \cdot \left(3 \cdot y\right)}}\right) \]

      10. *-commutative 96.3%

          \[\leadsto x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{z \cdot \color{blue}{\left(y \cdot 3\right)}}\right) \]

   3. Simplified 96.3%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{z \cdot \left(y \cdot 3\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.8%

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

   6. Taylor expanded in y around inf 95.7%

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

      1. associate-*r/ 95.7%

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

      2. *-commutative 95.7%

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

   8. Simplified 95.7%

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

   if -1.95e12 < y < 6.5e14

   1. Initial program 95.2%

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

      1. +-commutative 95.2%

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

      2. associate-+r- 95.2%

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

      3. +-commutative 95.2%

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

      4. associate--l+ 95.2%

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

      5. sub-neg 95.2%

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

      6. remove-double-neg 95.2%

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

      7. distribute-frac-neg 95.2%

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

      8. distribute-neg-in 95.2%

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

      9. remove-double-neg 95.2%

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

      10. sub-neg 95.2%

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

      11. neg-mul-1 95.2%

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

      12. times-frac 91.2%

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

      13. distribute-frac-neg 91.2%

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

      14. neg-mul-1 91.2%

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

      15. *-commutative 91.2%

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

      16. associate-/l* 91.2%

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

      17. *-commutative 91.2%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in t around inf 89.7%

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

   if 6.5e14 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0 95.2%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1950000000000:\\ \;\;\;\;x + \frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+14}:\\ \;\;\;\;x + 0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-158}:\\ \;\;\;\;x + \frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{\frac{1}{y}}{3}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -9e-158)
   (+ x (/ (* y -0.3333333333333333) z))
   (if (<= y 2.5e-116)
     (* (/ t z) (/ (/ 1.0 y) 3.0))
     (- x (* (/ y z) 0.3333333333333333)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9e-158) {
		tmp = x + ((y * -0.3333333333333333) / z);
	} else if (y <= 2.5e-116) {
		tmp = (t / z) * ((1.0 / y) / 3.0);
	} else {
		tmp = x - ((y / z) * 0.3333333333333333);
	}
	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 <= (-9d-158)) then
        tmp = x + ((y * (-0.3333333333333333d0)) / z)
    else if (y <= 2.5d-116) then
        tmp = (t / z) * ((1.0d0 / y) / 3.0d0)
    else
        tmp = x - ((y / z) * 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9e-158) {
		tmp = x + ((y * -0.3333333333333333) / z);
	} else if (y <= 2.5e-116) {
		tmp = (t / z) * ((1.0 / y) / 3.0);
	} else {
		tmp = x - ((y / z) * 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -9e-158:
		tmp = x + ((y * -0.3333333333333333) / z)
	elif y <= 2.5e-116:
		tmp = (t / z) * ((1.0 / y) / 3.0)
	else:
		tmp = x - ((y / z) * 0.3333333333333333)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9e-158)
		tmp = Float64(x + Float64(Float64(y * -0.3333333333333333) / z));
	elseif (y <= 2.5e-116)
		tmp = Float64(Float64(t / z) * Float64(Float64(1.0 / y) / 3.0));
	else
		tmp = Float64(x - Float64(Float64(y / z) * 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -9e-158)
		tmp = x + ((y * -0.3333333333333333) / z);
	elseif (y <= 2.5e-116)
		tmp = (t / z) * ((1.0 / y) / 3.0);
	else
		tmp = x - ((y / z) * 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -9e-158], N[(x + N[(N[(y * -0.3333333333333333), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.5e-116], N[(N[(t / z), $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y / z), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9e-158

   1. Initial program 96.6%

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

      1. sub-neg 96.6%

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

      2. associate-+l+ 96.6%

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

      3. distribute-frac-neg 96.6%

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

      4. neg-mul-1 96.6%

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

      5. *-commutative 96.6%

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

      6. times-frac 96.6%

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

      7. fma-define 96.6%

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

      8. metadata-eval 96.6%

         \[\leadsto x + \mathsf{fma}\left(\color{blue}{-0.3333333333333333}, \frac{y}{z}, \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      9. associate-*l* 96.7%

         \[\leadsto x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{\color{blue}{z \cdot \left(3 \cdot y\right)}}\right) \]

      10. *-commutative 96.7%

          \[\leadsto x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{z \cdot \color{blue}{\left(y \cdot 3\right)}}\right) \]

   3. Simplified 96.7%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{z \cdot \left(y \cdot 3\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 97.7%

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

   6. Taylor expanded in y around inf 84.4%

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

      1. associate-*r/ 84.4%

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

      2. *-commutative 84.4%

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

   8. Simplified 84.4%

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

   if -9e-158 < y < 2.5000000000000001e-116

   1. Initial program 92.5%

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

      1. +-commutative 92.5%

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

      2. associate-+r- 92.5%

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

      3. +-commutative 92.5%

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

      4. associate--l+ 92.5%

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

      5. sub-neg 92.5%

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

      6. remove-double-neg 92.5%

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

      7. distribute-frac-neg 92.5%

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

      8. distribute-neg-in 92.5%

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

      9. remove-double-neg 92.5%

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

      10. sub-neg 92.5%

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

      11. neg-mul-1 92.5%

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

      12. times-frac 90.2%

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

      13. distribute-frac-neg 90.2%

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

      14. neg-mul-1 90.2%

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

      15. *-commutative 90.2%

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

      16. associate-/l* 90.2%

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

      17. *-commutative 90.2%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in y around 0 97.2%

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

   6. Taylor expanded in t around inf 81.9%

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

      1. associate-*r/ 81.8%

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

      2. associate-*l/ 81.9%

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

      3. *-commutative 81.9%

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

   8. Simplified 81.9%

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

      1. *-commutative 81.9%

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

      2. associate-/l* 72.9%

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

      3. metadata-eval 72.9%

         \[\leadsto \frac{\color{blue}{\frac{1}{3}}}{z} \cdot \frac{t}{y} \]

      4. associate-/r* 72.9%

         \[\leadsto \color{blue}{\frac{1}{3 \cdot z}} \cdot \frac{t}{y} \]

      5. *-commutative 72.9%

         \[\leadsto \frac{1}{\color{blue}{z \cdot 3}} \cdot \frac{t}{y} \]

      6. associate-/r/ 72.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{z \cdot 3}{\frac{t}{y}}}} \]

      7. clear-num 72.9%

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

      8. div-inv 72.9%

         \[\leadsto \frac{\color{blue}{t \cdot \frac{1}{y}}}{z \cdot 3} \]

      9. times-frac 81.9%

         \[\leadsto \color{blue}{\frac{t}{z} \cdot \frac{\frac{1}{y}}{3}} \]

   10. Applied egg-rr 81.9%

       \[\leadsto \color{blue}{\frac{t}{z} \cdot \frac{\frac{1}{y}}{3}} \]

   if 2.5000000000000001e-116 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0 82.1%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-158}:\\ \;\;\;\;x + \frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{\frac{1}{y}}{3}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -5 \cdot 10^{+122}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \cdot 3 \leq 2 \cdot 10^{-63}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 3.0) -5e+122)
   x
   (if (<= (* z 3.0) 2e-63) (* -0.3333333333333333 (/ y z)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= -5e+122) {
		tmp = x;
	} else if ((z * 3.0) <= 2e-63) {
		tmp = -0.3333333333333333 * (y / 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 ((z * 3.0d0) <= (-5d+122)) then
        tmp = x
    else if ((z * 3.0d0) <= 2d-63) then
        tmp = (-0.3333333333333333d0) * (y / 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 ((z * 3.0) <= -5e+122) {
		tmp = x;
	} else if ((z * 3.0) <= 2e-63) {
		tmp = -0.3333333333333333 * (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * 3.0) <= -5e+122:
		tmp = x
	elif (z * 3.0) <= 2e-63:
		tmp = -0.3333333333333333 * (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 3.0) <= -5e+122)
		tmp = x;
	elseif (Float64(z * 3.0) <= 2e-63)
		tmp = Float64(-0.3333333333333333 * Float64(y / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * 3.0) <= -5e+122)
		tmp = x;
	elseif ((z * 3.0) <= 2e-63)
		tmp = -0.3333333333333333 * (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 3.0), $MachinePrecision], -5e+122], x, If[LessEqual[N[(z * 3.0), $MachinePrecision], 2e-63], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -4.99999999999999989e122 or 2.00000000000000013e-63 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 99.0%

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

      1. +-commutative 99.0%

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

      2. associate-+r- 99.0%

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

      3. +-commutative 99.0%

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

      4. associate--l+ 99.0%

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

      5. sub-neg 99.0%

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

      6. remove-double-neg 99.0%

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

      7. distribute-frac-neg 99.0%

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

      8. distribute-neg-in 99.0%

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

      9. remove-double-neg 99.0%

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

      10. sub-neg 99.0%

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

      11. neg-mul-1 99.0%

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

      12. times-frac 89.3%

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

      13. distribute-frac-neg 89.3%

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

      14. neg-mul-1 89.3%

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

      15. *-commutative 89.3%

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

      16. associate-/l* 89.3%

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

      17. *-commutative 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in x around inf 57.7%

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

   if -4.99999999999999989e122 < (*.f64 z #s(literal 3 binary64)) < 2.00000000000000013e-63

   1. Initial program 94.2%

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

   3. Taylor expanded in t around 0 56.2%

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

   4. Taylor expanded in x around 0 47.3%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{y}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.78 \cdot 10^{-158} \lor \neg \left(y \leq 4.6 \cdot 10^{-116}\right):\\ \;\;\;\;x + \frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.78e-158) (not (<= y 4.6e-116)))
   (+ x (/ (* y -0.3333333333333333) z))
   (* t (/ 0.3333333333333333 (* y z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.78e-158) || !(y <= 4.6e-116)) {
		tmp = x + ((y * -0.3333333333333333) / z);
	} else {
		tmp = t * (0.3333333333333333 / (y * 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 <= (-1.78d-158)) .or. (.not. (y <= 4.6d-116))) then
        tmp = x + ((y * (-0.3333333333333333d0)) / z)
    else
        tmp = t * (0.3333333333333333d0 / (y * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.78e-158) || !(y <= 4.6e-116)) {
		tmp = x + ((y * -0.3333333333333333) / z);
	} else {
		tmp = t * (0.3333333333333333 / (y * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.78e-158) or not (y <= 4.6e-116):
		tmp = x + ((y * -0.3333333333333333) / z)
	else:
		tmp = t * (0.3333333333333333 / (y * z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.78e-158) || !(y <= 4.6e-116))
		tmp = Float64(x + Float64(Float64(y * -0.3333333333333333) / z));
	else
		tmp = Float64(t * Float64(0.3333333333333333 / Float64(y * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.78e-158) || ~((y <= 4.6e-116)))
		tmp = x + ((y * -0.3333333333333333) / z);
	else
		tmp = t * (0.3333333333333333 / (y * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.78e-158], N[Not[LessEqual[y, 4.6e-116]], $MachinePrecision]], N[(x + N[(N[(y * -0.3333333333333333), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(t * N[(0.3333333333333333 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.77999999999999992e-158 or 4.60000000000000003e-116 < y

   1. Initial program 98.1%

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

      1. sub-neg 98.1%

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

      2. associate-+l+ 98.1%

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

      3. distribute-frac-neg 98.1%

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

      4. neg-mul-1 98.1%

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

      5. *-commutative 98.1%

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

      6. times-frac 98.2%

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

      7. fma-define 98.2%

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

      8. metadata-eval 98.2%

         \[\leadsto x + \mathsf{fma}\left(\color{blue}{-0.3333333333333333}, \frac{y}{z}, \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      9. associate-*l* 98.2%

         \[\leadsto x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{\color{blue}{z \cdot \left(3 \cdot y\right)}}\right) \]

      10. *-commutative 98.2%

          \[\leadsto x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{z \cdot \color{blue}{\left(y \cdot 3\right)}}\right) \]

   3. Simplified 98.2%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{z \cdot \left(y \cdot 3\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 97.1%

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

   6. Taylor expanded in y around inf 83.3%

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

      1. associate-*r/ 83.3%

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

      2. *-commutative 83.3%

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

   8. Simplified 83.3%

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

   if -1.77999999999999992e-158 < y < 4.60000000000000003e-116

   1. Initial program 92.5%

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

      1. +-commutative 92.5%

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

      2. associate-+r- 92.5%

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

      3. +-commutative 92.5%

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

      4. associate--l+ 92.5%

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

      5. sub-neg 92.5%

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

      6. remove-double-neg 92.5%

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

      7. distribute-frac-neg 92.5%

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

      8. distribute-neg-in 92.5%

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

      9. remove-double-neg 92.5%

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

      10. sub-neg 92.5%

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

      11. neg-mul-1 92.5%

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

      12. times-frac 90.2%

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

      13. distribute-frac-neg 90.2%

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

      14. neg-mul-1 90.2%

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

      15. *-commutative 90.2%

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

      16. associate-/l* 90.2%

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

      17. *-commutative 90.2%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in y around 0 97.2%

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

   6. Taylor expanded in t around inf 81.9%

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

      1. associate-*r/ 81.8%

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

      2. associate-*l/ 81.9%

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

      3. *-commutative 81.9%

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

   8. Simplified 81.9%

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

      1. associate-/l* 74.8%

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

      2. *-commutative 74.8%

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

      3. associate-/l/ 74.9%

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

   10. Applied egg-rr 74.9%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{y \cdot z} \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.78 \cdot 10^{-158} \lor \neg \left(y \leq 4.6 \cdot 10^{-116}\right):\\ \;\;\;\;x + \frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-158}:\\ \;\;\;\;x + \frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-116}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \frac{t}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -8.2e-158)
   (+ x (/ (* y -0.3333333333333333) z))
   (if (<= y 3e-116)
     (/ (* 0.3333333333333333 (/ t z)) y)
     (- x (* (/ y z) 0.3333333333333333)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -8.2e-158) {
		tmp = x + ((y * -0.3333333333333333) / z);
	} else if (y <= 3e-116) {
		tmp = (0.3333333333333333 * (t / z)) / y;
	} else {
		tmp = x - ((y / z) * 0.3333333333333333);
	}
	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 <= (-8.2d-158)) then
        tmp = x + ((y * (-0.3333333333333333d0)) / z)
    else if (y <= 3d-116) then
        tmp = (0.3333333333333333d0 * (t / z)) / y
    else
        tmp = x - ((y / z) * 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -8.2e-158) {
		tmp = x + ((y * -0.3333333333333333) / z);
	} else if (y <= 3e-116) {
		tmp = (0.3333333333333333 * (t / z)) / y;
	} else {
		tmp = x - ((y / z) * 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -8.2e-158:
		tmp = x + ((y * -0.3333333333333333) / z)
	elif y <= 3e-116:
		tmp = (0.3333333333333333 * (t / z)) / y
	else:
		tmp = x - ((y / z) * 0.3333333333333333)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -8.2e-158)
		tmp = Float64(x + Float64(Float64(y * -0.3333333333333333) / z));
	elseif (y <= 3e-116)
		tmp = Float64(Float64(0.3333333333333333 * Float64(t / z)) / y);
	else
		tmp = Float64(x - Float64(Float64(y / z) * 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -8.2e-158)
		tmp = x + ((y * -0.3333333333333333) / z);
	elseif (y <= 3e-116)
		tmp = (0.3333333333333333 * (t / z)) / y;
	else
		tmp = x - ((y / z) * 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -8.2e-158], N[(x + N[(N[(y * -0.3333333333333333), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3e-116], N[(N[(0.3333333333333333 * N[(t / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x - N[(N[(y / z), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.20000000000000008e-158

   1. Initial program 96.6%

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

      1. sub-neg 96.6%

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

      2. associate-+l+ 96.6%

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

      3. distribute-frac-neg 96.6%

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

      4. neg-mul-1 96.6%

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

      5. *-commutative 96.6%

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

      6. times-frac 96.6%

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

      7. fma-define 96.6%

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

      8. metadata-eval 96.6%

         \[\leadsto x + \mathsf{fma}\left(\color{blue}{-0.3333333333333333}, \frac{y}{z}, \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      9. associate-*l* 96.7%

         \[\leadsto x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{\color{blue}{z \cdot \left(3 \cdot y\right)}}\right) \]

      10. *-commutative 96.7%

          \[\leadsto x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{z \cdot \color{blue}{\left(y \cdot 3\right)}}\right) \]

   3. Simplified 96.7%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{z \cdot \left(y \cdot 3\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 97.7%

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

   6. Taylor expanded in y around inf 84.4%

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

      1. associate-*r/ 84.4%

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

      2. *-commutative 84.4%

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

   8. Simplified 84.4%

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

   if -8.20000000000000008e-158 < y < 3.00000000000000026e-116

   1. Initial program 92.5%

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

      1. +-commutative 92.5%

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

      2. associate-+r- 92.5%

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

      3. +-commutative 92.5%

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

      4. associate--l+ 92.5%

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

      5. sub-neg 92.5%

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

      6. remove-double-neg 92.5%

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

      7. distribute-frac-neg 92.5%

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

      8. distribute-neg-in 92.5%

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

      9. remove-double-neg 92.5%

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

      10. sub-neg 92.5%

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

      11. neg-mul-1 92.5%

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

      12. times-frac 90.2%

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

      13. distribute-frac-neg 90.2%

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

      14. neg-mul-1 90.2%

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

      15. *-commutative 90.2%

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

      16. associate-/l* 90.2%

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

      17. *-commutative 90.2%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in y around 0 97.2%

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

   6. Taylor expanded in t around inf 81.9%

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

   if 3.00000000000000026e-116 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0 82.1%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-158}:\\ \;\;\;\;x + \frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-116}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \frac{t}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-158}:\\ \;\;\;\;x + \frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-116}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -9e-158)
   (+ x (/ (* y -0.3333333333333333) z))
   (if (<= y 2.8e-116)
     (* t (/ 0.3333333333333333 (* y z)))
     (- x (* (/ y z) 0.3333333333333333)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9e-158) {
		tmp = x + ((y * -0.3333333333333333) / z);
	} else if (y <= 2.8e-116) {
		tmp = t * (0.3333333333333333 / (y * z));
	} else {
		tmp = x - ((y / z) * 0.3333333333333333);
	}
	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 <= (-9d-158)) then
        tmp = x + ((y * (-0.3333333333333333d0)) / z)
    else if (y <= 2.8d-116) then
        tmp = t * (0.3333333333333333d0 / (y * z))
    else
        tmp = x - ((y / z) * 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9e-158) {
		tmp = x + ((y * -0.3333333333333333) / z);
	} else if (y <= 2.8e-116) {
		tmp = t * (0.3333333333333333 / (y * z));
	} else {
		tmp = x - ((y / z) * 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -9e-158:
		tmp = x + ((y * -0.3333333333333333) / z)
	elif y <= 2.8e-116:
		tmp = t * (0.3333333333333333 / (y * z))
	else:
		tmp = x - ((y / z) * 0.3333333333333333)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9e-158)
		tmp = Float64(x + Float64(Float64(y * -0.3333333333333333) / z));
	elseif (y <= 2.8e-116)
		tmp = Float64(t * Float64(0.3333333333333333 / Float64(y * z)));
	else
		tmp = Float64(x - Float64(Float64(y / z) * 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -9e-158)
		tmp = x + ((y * -0.3333333333333333) / z);
	elseif (y <= 2.8e-116)
		tmp = t * (0.3333333333333333 / (y * z));
	else
		tmp = x - ((y / z) * 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -9e-158], N[(x + N[(N[(y * -0.3333333333333333), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e-116], N[(t * N[(0.3333333333333333 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y / z), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9e-158

   1. Initial program 96.6%

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

      1. sub-neg 96.6%

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

      2. associate-+l+ 96.6%

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

      3. distribute-frac-neg 96.6%

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

      4. neg-mul-1 96.6%

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

      5. *-commutative 96.6%

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

      6. times-frac 96.6%

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

      7. fma-define 96.6%

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

      8. metadata-eval 96.6%

         \[\leadsto x + \mathsf{fma}\left(\color{blue}{-0.3333333333333333}, \frac{y}{z}, \frac{t}{\left(z \cdot 3\right) \cdot y}\right) \]

      9. associate-*l* 96.7%

         \[\leadsto x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{\color{blue}{z \cdot \left(3 \cdot y\right)}}\right) \]

      10. *-commutative 96.7%

          \[\leadsto x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{z \cdot \color{blue}{\left(y \cdot 3\right)}}\right) \]

   3. Simplified 96.7%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, \frac{t}{z \cdot \left(y \cdot 3\right)}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 97.7%

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

   6. Taylor expanded in y around inf 84.4%

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

      1. associate-*r/ 84.4%

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

      2. *-commutative 84.4%

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

   8. Simplified 84.4%

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

   if -9e-158 < y < 2.7999999999999999e-116

   1. Initial program 92.5%

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

      1. +-commutative 92.5%

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

      2. associate-+r- 92.5%

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

      3. +-commutative 92.5%

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

      4. associate--l+ 92.5%

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

      5. sub-neg 92.5%

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

      6. remove-double-neg 92.5%

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

      7. distribute-frac-neg 92.5%

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

      8. distribute-neg-in 92.5%

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

      9. remove-double-neg 92.5%

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

      10. sub-neg 92.5%

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

      11. neg-mul-1 92.5%

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

      12. times-frac 90.2%

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

      13. distribute-frac-neg 90.2%

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

      14. neg-mul-1 90.2%

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

      15. *-commutative 90.2%

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

      16. associate-/l* 90.2%

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

      17. *-commutative 90.2%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in y around 0 97.2%

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

   6. Taylor expanded in t around inf 81.9%

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

      1. associate-*r/ 81.8%

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

      2. associate-*l/ 81.9%

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

      3. *-commutative 81.9%

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

   8. Simplified 81.9%

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

      1. associate-/l* 74.8%

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

      2. *-commutative 74.8%

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

      3. associate-/l/ 74.9%

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

   10. Applied egg-rr 74.9%

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

   if 2.7999999999999999e-116 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0 82.1%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-158}:\\ \;\;\;\;x + \frac{y \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-116}:\\ \;\;\;\;t \cdot \frac{0.3333333333333333}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z} \cdot 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.8% 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 96.4%

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

   1. +-commutative 96.4%

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

   2. associate-+r- 96.4%

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

   3. +-commutative 96.4%

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

   4. associate--l+ 96.4%

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

   5. sub-neg 96.4%

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

   6. remove-double-neg 96.4%

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

   7. distribute-frac-neg 96.4%

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

   8. distribute-neg-in 96.4%

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

   9. remove-double-neg 96.4%

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

   10. sub-neg 96.4%

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

   11. neg-mul-1 96.4%

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

   12. times-frac 94.2%

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

   13. distribute-frac-neg 94.2%

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

   14. neg-mul-1 94.2%

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

   15. *-commutative 94.2%

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

   16. associate-/l* 94.2%

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

   17. *-commutative 94.2%

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

3. Simplified 95.0%

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

Alternative 13: 30.4% 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 96.4%

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

   1. +-commutative 96.4%

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

   2. associate-+r- 96.4%

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

   3. +-commutative 96.4%

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

   4. associate--l+ 96.4%

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

   5. sub-neg 96.4%

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

   6. remove-double-neg 96.4%

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

   7. distribute-frac-neg 96.4%

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

   8. distribute-neg-in 96.4%

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

   9. remove-double-neg 96.4%

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

   10. sub-neg 96.4%

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

   11. neg-mul-1 96.4%

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

   12. times-frac 94.2%

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

   13. distribute-frac-neg 94.2%

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

   14. neg-mul-1 94.2%

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

   15. *-commutative 94.2%

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

   16. associate-/l* 94.2%

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

   17. *-commutative 94.2%

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

3. Simplified 95.0%

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

5. Taylor expanded in x around inf 32.2%

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

Developer target: 96.2% 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 2024101 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

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

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