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

Percentage Accurate: 95.5% → 99.5%

Time: 12.0s

Alternatives: 18

Speedup: 1.4×

• 28.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{-37}:\\ \;\;\;\;t\_1 + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{elif}\;z \cdot 3 \leq 2 \cdot 10^{+43}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{y} - y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y (* z 3.0)))))
   (if (<= (* z 3.0) -1e-37)
     (+ t_1 (/ t (* z (* y 3.0))))
     (if (<= (* z 3.0) 2e+43)
       (+ x (/ (/ (- (/ t y) y) z) 3.0))
       (+ t_1 (/ t (* y (* z 3.0))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (z * 3.0));
	double tmp;
	if ((z * 3.0) <= -1e-37) {
		tmp = t_1 + (t / (z * (y * 3.0)));
	} else if ((z * 3.0) <= 2e+43) {
		tmp = x + ((((t / y) - y) / z) / 3.0);
	} else {
		tmp = t_1 + (t / (y * (z * 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x - (y / (z * 3.0d0))
    if ((z * 3.0d0) <= (-1d-37)) then
        tmp = t_1 + (t / (z * (y * 3.0d0)))
    else if ((z * 3.0d0) <= 2d+43) then
        tmp = x + ((((t / y) - y) / z) / 3.0d0)
    else
        tmp = t_1 + (t / (y * (z * 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (z * 3.0));
	double tmp;
	if ((z * 3.0) <= -1e-37) {
		tmp = t_1 + (t / (z * (y * 3.0)));
	} else if ((z * 3.0) <= 2e+43) {
		tmp = x + ((((t / y) - y) / z) / 3.0);
	} else {
		tmp = t_1 + (t / (y * (z * 3.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / (z * 3.0))
	tmp = 0
	if (z * 3.0) <= -1e-37:
		tmp = t_1 + (t / (z * (y * 3.0)))
	elif (z * 3.0) <= 2e+43:
		tmp = x + ((((t / y) - y) / z) / 3.0)
	else:
		tmp = t_1 + (t / (y * (z * 3.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / Float64(z * 3.0)))
	tmp = 0.0
	if (Float64(z * 3.0) <= -1e-37)
		tmp = Float64(t_1 + Float64(t / Float64(z * Float64(y * 3.0))));
	elseif (Float64(z * 3.0) <= 2e+43)
		tmp = Float64(x + Float64(Float64(Float64(Float64(t / y) - y) / z) / 3.0));
	else
		tmp = Float64(t_1 + Float64(t / Float64(y * Float64(z * 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / (z * 3.0));
	tmp = 0.0;
	if ((z * 3.0) <= -1e-37)
		tmp = t_1 + (t / (z * (y * 3.0)));
	elseif ((z * 3.0) <= 2e+43)
		tmp = x + ((((t / y) - y) / z) / 3.0);
	else
		tmp = t_1 + (t / (y * (z * 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * 3.0), $MachinePrecision], -1e-37], N[(t$95$1 + N[(t / N[(z * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * 3.0), $MachinePrecision], 2e+43], N[(x + N[(N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -1.00000000000000007e-37

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

   if -1.00000000000000007e-37 < (*.f64 z #s(literal 3 binary64)) < 2.00000000000000003e43

   1. Initial program 89.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.9

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

   6. Applied rewrites 99.9%

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

   if 2.00000000000000003e43 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{-37}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{elif}\;z \cdot 3 \leq 2 \cdot 10^{+43}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{y} - y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;t\_1 + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 1.5 \cdot 10^{+303}:\\ \;\;\;\;t\_1 + \frac{\frac{t}{z}}{y \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{1}{z} \cdot \left(\frac{t}{y} - y\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y (* z 3.0)))))
   (if (<= (+ t_1 (/ t (* y (* z 3.0)))) 1.5e+303)
     (+ t_1 (/ (/ t z) (* y 3.0)))
     (fma 0.3333333333333333 (* (/ 1.0 z) (- (/ t y) y)) x))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / (z * 3.0));
	double tmp;
	if ((t_1 + (t / (y * (z * 3.0)))) <= 1.5e+303) {
		tmp = t_1 + ((t / z) / (y * 3.0));
	} else {
		tmp = fma(0.3333333333333333, ((1.0 / z) * ((t / y) - y)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / Float64(z * 3.0)))
	tmp = 0.0
	if (Float64(t_1 + Float64(t / Float64(y * Float64(z * 3.0)))) <= 1.5e+303)
		tmp = Float64(t_1 + Float64(Float64(t / z) / Float64(y * 3.0)));
	else
		tmp = fma(0.3333333333333333, Float64(Float64(1.0 / z) * Float64(Float64(t / y) - y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5e+303], N[(t$95$1 + N[(N[(t / z), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(N[(1.0 / z), $MachinePrecision] * N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] + x), $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))) < 1.49999999999999985e303

   1. Initial program 97.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.9

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

   4. Applied rewrites 98.9%

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

   if 1.49999999999999985e303 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 81.4%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. distribute-lft-out-- N/A

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

      13. lower-fma.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(0.3333333333333333, \color{blue}{\frac{1}{z} \cdot \left(\frac{t}{y} - y\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)} \leq 1.5 \cdot 10^{+303}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z}}{y \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{1}{z} \cdot \left(\frac{t}{y} - y\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{-37}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{elif}\;z \cdot 3 \leq 2 \cdot 10^{+43}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{y} - y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right) - \frac{t}{\left(y \cdot z\right) \cdot -3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z 3.0) -1e-37)
   (+ (- x (/ y (* z 3.0))) (/ t (* z (* y 3.0))))
   (if (<= (* z 3.0) 2e+43)
     (+ x (/ (/ (- (/ t y) y) z) 3.0))
     (- (fma (/ y z) -0.3333333333333333 x) (/ t (* (* y z) -3.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * 3.0) <= -1e-37) {
		tmp = (x - (y / (z * 3.0))) + (t / (z * (y * 3.0)));
	} else if ((z * 3.0) <= 2e+43) {
		tmp = x + ((((t / y) - y) / z) / 3.0);
	} else {
		tmp = fma((y / z), -0.3333333333333333, x) - (t / ((y * z) * -3.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * 3.0) <= -1e-37)
		tmp = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(z * Float64(y * 3.0))));
	elseif (Float64(z * 3.0) <= 2e+43)
		tmp = Float64(x + Float64(Float64(Float64(Float64(t / y) - y) / z) / 3.0));
	else
		tmp = Float64(fma(Float64(y / z), -0.3333333333333333, x) - Float64(t / Float64(Float64(y * z) * -3.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * 3.0), $MachinePrecision], -1e-37], N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(z * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * 3.0), $MachinePrecision], 2e+43], N[(x + N[(N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision] - N[(t / N[(N[(y * z), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -1.00000000000000007e-37

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

   if -1.00000000000000007e-37 < (*.f64 z #s(literal 3 binary64)) < 2.00000000000000003e43

   1. Initial program 89.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.9

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

   6. Applied rewrites 99.9%

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

   if 2.00000000000000003e43 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. div-inv N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lift-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{-37}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{elif}\;z \cdot 3 \leq 2 \cdot 10^{+43}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{y} - y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right) - \frac{t}{\left(y \cdot z\right) \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right) - \frac{t}{\left(y \cdot z\right) \cdot -3}\\ \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot 3 \leq 2 \cdot 10^{+43}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{y} - y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (fma (/ y z) -0.3333333333333333 x) (/ t (* (* y z) -3.0)))))
   (if (<= (* z 3.0) -1e+41)
     t_1
     (if (<= (* z 3.0) 2e+43) (+ x (/ (/ (- (/ t y) y) z) 3.0)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / z), -0.3333333333333333, x) - (t / ((y * z) * -3.0));
	double tmp;
	if ((z * 3.0) <= -1e+41) {
		tmp = t_1;
	} else if ((z * 3.0) <= 2e+43) {
		tmp = x + ((((t / y) - y) / z) / 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(Float64(y / z), -0.3333333333333333, x) - Float64(t / Float64(Float64(y * z) * -3.0)))
	tmp = 0.0
	if (Float64(z * 3.0) <= -1e+41)
		tmp = t_1;
	elseif (Float64(z * 3.0) <= 2e+43)
		tmp = Float64(x + Float64(Float64(Float64(Float64(t / y) - y) / z) / 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision] - N[(t / N[(N[(y * z), $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * 3.0), $MachinePrecision], -1e+41], t$95$1, If[LessEqual[N[(z * 3.0), $MachinePrecision], 2e+43], N[(x + N[(N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -1.00000000000000001e41 or 2.00000000000000003e43 < (*.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. div-inv N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lift-/.f64 N/A

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

   4. Applied rewrites 99.0%

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

   if -1.00000000000000001e41 < (*.f64 z #s(literal 3 binary64)) < 2.00000000000000003e43

   1. Initial program 90.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.9

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right) - \frac{t}{\left(y \cdot z\right) \cdot -3}\\ \mathbf{elif}\;z \cdot 3 \leq 2 \cdot 10^{+43}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{y} - y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right) - \frac{t}{\left(y \cdot z\right) \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\right)\\ \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot 3 \leq 10^{+34}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{y} - y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1
         (fma
          (/ t (* y z))
          0.3333333333333333
          (fma y (/ -0.3333333333333333 z) x))))
   (if (<= (* z 3.0) -1e+41)
     t_1
     (if (<= (* z 3.0) 1e+34) (+ x (/ (/ (- (/ t y) y) z) 3.0)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((t / (y * z)), 0.3333333333333333, fma(y, (-0.3333333333333333 / z), x));
	double tmp;
	if ((z * 3.0) <= -1e+41) {
		tmp = t_1;
	} else if ((z * 3.0) <= 1e+34) {
		tmp = x + ((((t / y) - y) / z) / 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(t / Float64(y * z)), 0.3333333333333333, fma(y, Float64(-0.3333333333333333 / z), x))
	tmp = 0.0
	if (Float64(z * 3.0) <= -1e+41)
		tmp = t_1;
	elseif (Float64(z * 3.0) <= 1e+34)
		tmp = Float64(x + Float64(Float64(Float64(Float64(t / y) - y) / z) / 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * 3.0), $MachinePrecision], -1e+41], t$95$1, If[LessEqual[N[(z * 3.0), $MachinePrecision], 1e+34], N[(x + N[(N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -1.00000000000000001e41 or 9.99999999999999946e33 < (*.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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 99.0

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

      14. lift--.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. div-inv N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-/r* N/A

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

      20. metadata-eval N/A

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

      21. associate-/l* N/A

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

      22. lift-*.f64 N/A

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

      23. lift-/.f64 N/A

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

   6. Applied rewrites 99.0%

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

   if -1.00000000000000001e41 < (*.f64 z #s(literal 3 binary64)) < 9.99999999999999946e33

   1. Initial program 90.3%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.9

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\right)\\ \mathbf{elif}\;z \cdot 3 \leq 10^{+34}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{y} - y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\\ \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, t\_1\right)\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{+66}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{y} - y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{y \cdot z}, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y z) -0.3333333333333333 x)))
   (if (<= (* z 3.0) -1e+41)
     (fma (/ t (* y z)) 0.3333333333333333 t_1)
     (if (<= (* z 3.0) 5e+66)
       (+ x (/ (/ (- (/ t y) y) z) 3.0))
       (fma t (/ 0.3333333333333333 (* y z)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / z), -0.3333333333333333, x);
	double tmp;
	if ((z * 3.0) <= -1e+41) {
		tmp = fma((t / (y * z)), 0.3333333333333333, t_1);
	} else if ((z * 3.0) <= 5e+66) {
		tmp = x + ((((t / y) - y) / z) / 3.0);
	} else {
		tmp = fma(t, (0.3333333333333333 / (y * z)), t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / z), -0.3333333333333333, x)
	tmp = 0.0
	if (Float64(z * 3.0) <= -1e+41)
		tmp = fma(Float64(t / Float64(y * z)), 0.3333333333333333, t_1);
	elseif (Float64(z * 3.0) <= 5e+66)
		tmp = Float64(x + Float64(Float64(Float64(Float64(t / y) - y) / z) / 3.0));
	else
		tmp = fma(t, Float64(0.3333333333333333 / Float64(y * z)), t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]}, If[LessEqual[N[(z * 3.0), $MachinePrecision], -1e+41], N[(N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + t$95$1), $MachinePrecision], If[LessEqual[N[(z * 3.0), $MachinePrecision], 5e+66], N[(x + N[(N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(0.3333333333333333 / N[(y * z), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -1.00000000000000001e41

   1. Initial program 98.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 98.3

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

      14. lift--.f64 N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. lift-/.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-/r* N/A

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

      20. div-inv N/A

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

      21. distribute-rgt-neg-in N/A

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

      22. lower-fma.f64 N/A

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

   4. Applied rewrites 98.2%

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

   if -1.00000000000000001e41 < (*.f64 z #s(literal 3 binary64)) < 4.99999999999999991e66

   1. Initial program 90.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.9

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

   6. Applied rewrites 99.9%

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

   if 4.99999999999999991e66 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 88.7

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

   4. Applied rewrites 88.7%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 88.7

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

   6. Applied rewrites 88.7%

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. associate--r- N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. associate-/r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

   8. Applied rewrites 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{+66}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{y} - y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{y \cdot z}, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{0.3333333333333333}{y \cdot z}, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)\\ \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{+66}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{y} - y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1
         (fma
          t
          (/ 0.3333333333333333 (* y z))
          (fma (/ y z) -0.3333333333333333 x))))
   (if (<= (* z 3.0) -1e+132)
     t_1
     (if (<= (* z 3.0) 5e+66) (+ x (/ (/ (- (/ t y) y) z) 3.0)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(t, (0.3333333333333333 / (y * z)), fma((y / z), -0.3333333333333333, x));
	double tmp;
	if ((z * 3.0) <= -1e+132) {
		tmp = t_1;
	} else if ((z * 3.0) <= 5e+66) {
		tmp = x + ((((t / y) - y) / z) / 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(t, Float64(0.3333333333333333 / Float64(y * z)), fma(Float64(y / z), -0.3333333333333333, x))
	tmp = 0.0
	if (Float64(z * 3.0) <= -1e+132)
		tmp = t_1;
	elseif (Float64(z * 3.0) <= 5e+66)
		tmp = Float64(x + Float64(Float64(Float64(Float64(t / y) - y) / z) / 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * N[(0.3333333333333333 / N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(N[(y / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * 3.0), $MachinePrecision], -1e+132], t$95$1, If[LessEqual[N[(z * 3.0), $MachinePrecision], 5e+66], N[(x + N[(N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -9.99999999999999991e131 or 4.99999999999999991e66 < (*.f64 z #s(literal 3 binary64))

   1. Initial program 98.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 85.6

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

   4. Applied rewrites 85.6%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 85.6

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

   6. Applied rewrites 85.6%

      \[\leadsto x - \color{blue}{\frac{\frac{y - \frac{t}{y}}{3}}{z}} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. associate--r- N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. div-inv N/A

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. associate-/r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

   8. Applied rewrites 98.6%

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

   if -9.99999999999999991e131 < (*.f64 z #s(literal 3 binary64)) < 4.99999999999999991e66

   1. Initial program 92.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.9

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{y \cdot z}, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{+66}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{y} - y}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{y \cdot z}, \mathsf{fma}\left(\frac{y}{z}, -0.3333333333333333, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\frac{\frac{t}{y} - y}{3}}{z}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{-92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \frac{t}{z}, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (/ (- (/ t y) y) 3.0) z))))
   (if (<= y -1.6e-92)
     t_1
     (if (<= y 9.6e-76) (/ (fma 0.3333333333333333 (/ t z) (* x y)) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + ((((t / y) - y) / 3.0) / z);
	double tmp;
	if (y <= -1.6e-92) {
		tmp = t_1;
	} else if (y <= 9.6e-76) {
		tmp = fma(0.3333333333333333, (t / z), (x * y)) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(Float64(Float64(t / y) - y) / 3.0) / z))
	tmp = 0.0
	if (y <= -1.6e-92)
		tmp = t_1;
	elseif (y <= 9.6e-76)
		tmp = Float64(fma(0.3333333333333333, Float64(t / z), Float64(x * y)) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / 3.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6e-92], t$95$1, If[LessEqual[y, 9.6e-76], N[(N[(0.3333333333333333 * N[(t / z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5999999999999998e-92 or 9.60000000000000053e-76 < y

   1. Initial program 95.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{x - \frac{y - \frac{t}{y}}{z \cdot 3}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if -1.5999999999999998e-92 < y < 9.60000000000000053e-76

   1. Initial program 93.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z} + x \cdot y}{y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.5

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

   5. Applied rewrites 95.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \frac{t}{z}, y \cdot x\right)}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-92}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{y} - y}{3}}{z}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \frac{t}{z}, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\frac{t}{y} - y}{3}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \frac{t}{z}, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (- (/ t y) y) (* z 3.0)))))
   (if (<= y -2.3e-64)
     t_1
     (if (<= y 9.6e-76) (/ (fma 0.3333333333333333 (/ t z) (* x y)) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (((t / y) - y) / (z * 3.0));
	double tmp;
	if (y <= -2.3e-64) {
		tmp = t_1;
	} else if (y <= 9.6e-76) {
		tmp = fma(0.3333333333333333, (t / z), (x * y)) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(Float64(t / y) - y) / Float64(z * 3.0)))
	tmp = 0.0
	if (y <= -2.3e-64)
		tmp = t_1;
	elseif (y <= 9.6e-76)
		tmp = Float64(fma(0.3333333333333333, Float64(t / z), Float64(x * y)) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e-64], t$95$1, If[LessEqual[y, 9.6e-76], N[(N[(0.3333333333333333 * N[(t / z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.3000000000000001e-64 or 9.60000000000000053e-76 < y

   1. Initial program 95.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if -2.3000000000000001e-64 < y < 9.60000000000000053e-76

   1. Initial program 92.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z} + x \cdot y}{y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.7

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

   5. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \frac{t}{z}, y \cdot x\right)}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-64}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \frac{t}{z}, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t}{y} - y}{z \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{y} - y\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{-92}:\\ \;\;\;\;x - \frac{t\_1 \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \frac{t}{z}, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z}, t\_1, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ t y) y)))
   (if (<= y -1.5e-92)
     (- x (/ (* t_1 -0.3333333333333333) z))
     (if (<= y 9.6e-76)
       (/ (fma 0.3333333333333333 (/ t z) (* x y)) y)
       (fma (/ 0.3333333333333333 z) t_1 x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t / y) - y;
	double tmp;
	if (y <= -1.5e-92) {
		tmp = x - ((t_1 * -0.3333333333333333) / z);
	} else if (y <= 9.6e-76) {
		tmp = fma(0.3333333333333333, (t / z), (x * y)) / y;
	} else {
		tmp = fma((0.3333333333333333 / z), t_1, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t / y) - y)
	tmp = 0.0
	if (y <= -1.5e-92)
		tmp = Float64(x - Float64(Float64(t_1 * -0.3333333333333333) / z));
	elseif (y <= 9.6e-76)
		tmp = Float64(fma(0.3333333333333333, Float64(t / z), Float64(x * y)) / y);
	else
		tmp = fma(Float64(0.3333333333333333 / z), t_1, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[y, -1.5e-92], N[(x - N[(N[(t$95$1 * -0.3333333333333333), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.6e-76], N[(N[(0.3333333333333333 * N[(t / z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(0.3333333333333333 / z), $MachinePrecision] * t$95$1 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.50000000000000007e-92

   1. Initial program 97.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. distribute-rgt-in N/A

         \[\leadsto x - \color{blue}{\left(\left(\frac{-1}{3} \cdot \frac{t}{{y}^{2} \cdot z}\right) \cdot y + \left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto x - \left(\color{blue}{y \cdot \left(\frac{-1}{3} \cdot \frac{t}{{y}^{2} \cdot z}\right)} + \left(\frac{1}{3} \cdot \frac{1}{z}\right) \cdot y\right) \]

      3. associate-*l* N/A

         \[\leadsto x - \left(y \cdot \left(\frac{-1}{3} \cdot \frac{t}{{y}^{2} \cdot z}\right) + \color{blue}{\frac{1}{3} \cdot \left(\frac{1}{z} \cdot y\right)}\right) \]

      4. associate-*l/ N/A

         \[\leadsto x - \left(y \cdot \left(\frac{-1}{3} \cdot \frac{t}{{y}^{2} \cdot z}\right) + \frac{1}{3} \cdot \color{blue}{\frac{1 \cdot y}{z}}\right) \]

      5. *-lft-identity N/A

         \[\leadsto x - \left(y \cdot \left(\frac{-1}{3} \cdot \frac{t}{{y}^{2} \cdot z}\right) + \frac{1}{3} \cdot \frac{\color{blue}{y}}{z}\right) \]

      6. associate-*r/ N/A

         \[\leadsto x - \left(y \cdot \color{blue}{\frac{\frac{-1}{3} \cdot t}{{y}^{2} \cdot z}} + \frac{1}{3} \cdot \frac{y}{z}\right) \]

      7. associate-*r/ N/A

         \[\leadsto x - \left(\color{blue}{\frac{y \cdot \left(\frac{-1}{3} \cdot t\right)}{{y}^{2} \cdot z}} + \frac{1}{3} \cdot \frac{y}{z}\right) \]

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. times-frac N/A

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

      11. *-inverses N/A

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

      12. associate-/r* N/A

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

      13. associate-*r/ N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

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

      16. cancel-sign-sub-inv N/A

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

      17. associate-*r/ N/A

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

   7. Applied rewrites 99.7%

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

   if -1.50000000000000007e-92 < y < 9.60000000000000053e-76

   1. Initial program 93.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z} + x \cdot y}{y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.5

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

   5. Applied rewrites 95.5%

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

   if 9.60000000000000053e-76 < y

   1. Initial program 92.8%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. distribute-lft-out-- N/A

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

      13. lower-fma.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-92}:\\ \;\;\;\;x - \frac{\left(\frac{t}{y} - y\right) \cdot -0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \frac{t}{z}, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \mathbf{if}\;y \leq -5.1 \cdot 10^{-64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \frac{t}{z}, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ 0.3333333333333333 z) (- (/ t y) y) x)))
   (if (<= y -5.1e-64)
     t_1
     (if (<= y 9.6e-76) (/ (fma 0.3333333333333333 (/ t z) (* x y)) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((0.3333333333333333 / z), ((t / y) - y), x);
	double tmp;
	if (y <= -5.1e-64) {
		tmp = t_1;
	} else if (y <= 9.6e-76) {
		tmp = fma(0.3333333333333333, (t / z), (x * y)) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(0.3333333333333333 / z), Float64(Float64(t / y) - y), x)
	tmp = 0.0
	if (y <= -5.1e-64)
		tmp = t_1;
	elseif (y <= 9.6e-76)
		tmp = Float64(fma(0.3333333333333333, Float64(t / z), Float64(x * y)) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(0.3333333333333333 / z), $MachinePrecision] * N[(N[(t / y), $MachinePrecision] - y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -5.1e-64], t$95$1, If[LessEqual[y, 9.6e-76], N[(N[(0.3333333333333333 * N[(t / z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.09999999999999984e-64 or 9.60000000000000053e-76 < y

   1. Initial program 95.5%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. associate-*l/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. distribute-lft-out-- N/A

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

      13. lower-fma.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -5.09999999999999984e-64 < y < 9.60000000000000053e-76

   1. Initial program 92.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \frac{t}{z} + x \cdot y}{y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.7

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

   5. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.3333333333333333, \frac{t}{z}, y \cdot x\right)}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.3333333333333333, \frac{t}{z}, x \cdot y\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.3333333333333333}{z}, \frac{t}{y} - y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{if}\;y \leq -5.1 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 29000:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333, \frac{t}{y \cdot z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (/ -0.3333333333333333 z) x)))
   (if (<= y -5.1e+42)
     t_1
     (if (<= y 29000.0) (fma 0.3333333333333333 (/ t (* y z)) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (-0.3333333333333333 / z), x);
	double tmp;
	if (y <= -5.1e+42) {
		tmp = t_1;
	} else if (y <= 29000.0) {
		tmp = fma(0.3333333333333333, (t / (y * z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(-0.3333333333333333 / z), x)
	tmp = 0.0
	if (y <= -5.1e+42)
		tmp = t_1;
	elseif (y <= 29000.0)
		tmp = fma(0.3333333333333333, Float64(t / Float64(y * z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -5.1e+42], t$95$1, If[LessEqual[y, 29000.0], N[(0.3333333333333333 * N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.0999999999999999e42 or 29000 < y

   1. Initial program 96.0%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

   5. Applied rewrites 91.4%

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

   if -5.0999999999999999e42 < y < 29000

   1. Initial program 93.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x \cdot y - \frac{-1}{3} \cdot \frac{t}{z}}{y}} \]
   6. Step-by-step derivation

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. remove-double-neg N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. associate-/l/ N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. associate-*l/ N/A

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

      15. associate-/l* N/A

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

      16. *-inverses N/A

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

      17. *-rgt-identity N/A

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

      18. mul-1-neg N/A

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

   7. Applied rewrites 86.0%

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

Alternative 13: 75.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{if}\;y \leq -1.32 \cdot 10^{-92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 17000:\\ \;\;\;\;\frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (/ -0.3333333333333333 z) x)))
   (if (<= y -1.32e-92) t_1 (if (<= y 17000.0) (/ t (* y (* z 3.0))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (-0.3333333333333333 / z), x);
	double tmp;
	if (y <= -1.32e-92) {
		tmp = t_1;
	} else if (y <= 17000.0) {
		tmp = t / (y * (z * 3.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(-0.3333333333333333 / z), x)
	tmp = 0.0
	if (y <= -1.32e-92)
		tmp = t_1;
	elseif (y <= 17000.0)
		tmp = Float64(t / Float64(y * Float64(z * 3.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.32e-92], t$95$1, If[LessEqual[y, 17000.0], N[(t / N[(y * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3200000000000001e-92 or 17000 < y

   1. Initial program 95.9%

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

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*l/ N/A

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

      9. associate-*r/ N/A

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

      10. cancel-sign-sub N/A

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

      11. mul-1-neg N/A

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

      12. associate-*r/ N/A

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

      13. associate-*l/ N/A

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

      14. associate-/l* N/A

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

      15. mul-1-neg N/A

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

      16. *-inverses N/A

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

      17. cancel-sign-sub N/A

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

      18. *-rgt-identity N/A

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

   5. Applied rewrites 87.9%

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

   if -1.3200000000000001e-92 < y < 17000

   1. Initial program 92.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 64.9

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

   5. Applied rewrites 64.9%

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

   7. Applied rewrites 65.1%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 17000:\\ \;\;\;\;\frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 95.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. associate-*r/ N/A

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

   9. associate-*l/ N/A

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

   10. metadata-eval N/A

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

   11. associate-*r/ N/A

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

   12. distribute-lft-out-- N/A

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

   13. lower-fma.f64 N/A

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

   14. associate-*r/ N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 N/A

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

   17. lower--.f64 N/A

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

   18. lower-/.f64 94.6

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

5. Applied rewrites 94.6%

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

Alternative 15: 64.1% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. lower--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

   10. sub-div N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-/.f64 94.6

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

4. Applied rewrites 94.6%

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

5. Taylor expanded in y around inf

   \[\leadsto x - \color{blue}{\frac{1}{3} \cdot \frac{y}{z}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 64.6

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

7. Applied rewrites 64.6%

   \[\leadsto x - \color{blue}{\frac{y \cdot 0.3333333333333333}{z}} \]
8. Add Preprocessing

Alternative 16: 64.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, \frac{-0.3333333333333333}{z}, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in y around inf

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. associate-*l/ N/A

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

   9. associate-*r/ N/A

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

   10. cancel-sign-sub N/A

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

   11. mul-1-neg N/A

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

   12. associate-*r/ N/A

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

   13. associate-*l/ N/A

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

   14. associate-/l* N/A

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

   15. mul-1-neg N/A

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

   16. *-inverses N/A

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

   17. cancel-sign-sub N/A

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

   18. *-rgt-identity N/A

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

5. Applied rewrites 64.6%

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

Alternative 17: 36.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 34.5

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

5. Applied rewrites 34.5%

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

7. Applied rewrites 34.6%

   \[\leadsto \color{blue}{\frac{y}{z \cdot -3}} \]
8. Add Preprocessing

Alternative 18: 36.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{y}{z}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 34.5

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

5. Applied rewrites 34.5%

   \[\leadsto \color{blue}{y \cdot \frac{-0.3333333333333333}{z}} \]
6. Add Preprocessing

Developer Target 1: 95.8% accurate, 0.9× 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 2024222 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (- x (/ y (* z 3))) (/ (/ t (* z 3)) y)))

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