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

Percentage Accurate: 95.8% → 97.2%

Time: 9.5s

Alternatives: 18

Speedup: 1.4×

• 27.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.2% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ t (* (* 3.0 z) y))))
   (if (<= (+ t_1 (- x (/ y (* 3.0 z)))) (- INFINITY))
     (- x (/ (- y (/ t y)) (* 3.0 z)))
     (+ (- x (* (/ y -3.0) (/ -1.0 z))) t_1))))

C

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

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = t / ((3.0 * z) * y);
	double tmp;
	if ((t_1 + (x - (y / (3.0 * z)))) <= -Double.POSITIVE_INFINITY) {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	} else {
		tmp = (x - ((y / -3.0) * (-1.0 / z))) + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = t / ((3.0 * z) * y)
	tmp = 0
	if (t_1 + (x - (y / (3.0 * z)))) <= -math.inf:
		tmp = x - ((y - (t / y)) / (3.0 * z))
	else:
		tmp = (x - ((y / -3.0) * (-1.0 / z))) + t_1
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = t / ((3.0 * z) * y);
	tmp = 0.0;
	if ((t_1 + (x - (y / (3.0 * z)))) <= -Inf)
		tmp = x - ((y - (t / y)) / (3.0 * z));
	else
		tmp = (x - ((y / -3.0) * (-1.0 / z))) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(N[(y / -3.0), $MachinePrecision] * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $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))) < -inf.0

   1. Initial program 86.4%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 100.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if -inf.0 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 98.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 98.1

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

   4. Applied rewrites 98.1%

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

4. Final simplification 98.3%

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

Alternative 2: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\left(3 \cdot z\right) \cdot y}\\ \mathbf{if}\;t\_1 + \left(x - \frac{y}{3 \cdot z}\right) \leq -5 \cdot 10^{+269}:\\ \;\;\;\;x - \frac{\frac{y - \frac{t}{y}}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{0.3333333333333333 \cdot y}{z}\right) + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ t (* (* 3.0 z) y))))
   (if (<= (+ t_1 (- x (/ y (* 3.0 z)))) -5e+269)
     (- x (/ (/ (- y (/ t y)) z) 3.0))
     (+ (- x (/ (* 0.3333333333333333 y) z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = t / ((3.0 * z) * y);
	double tmp;
	if ((t_1 + (x - (y / (3.0 * z)))) <= -5e+269) {
		tmp = x - (((y - (t / y)) / z) / 3.0);
	} else {
		tmp = (x - ((0.3333333333333333 * y) / z)) + t_1;
	}
	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 = t / ((3.0d0 * z) * y)
    if ((t_1 + (x - (y / (3.0d0 * z)))) <= (-5d+269)) then
        tmp = x - (((y - (t / y)) / z) / 3.0d0)
    else
        tmp = (x - ((0.3333333333333333d0 * y) / z)) + t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = t / ((3.0 * z) * y);
	double tmp;
	if ((t_1 + (x - (y / (3.0 * z)))) <= -5e+269) {
		tmp = x - (((y - (t / y)) / z) / 3.0);
	} else {
		tmp = (x - ((0.3333333333333333 * y) / z)) + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = t / ((3.0 * z) * y)
	tmp = 0
	if (t_1 + (x - (y / (3.0 * z)))) <= -5e+269:
		tmp = x - (((y - (t / y)) / z) / 3.0)
	else:
		tmp = (x - ((0.3333333333333333 * y) / z)) + t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(t / Float64(Float64(3.0 * z) * y))
	tmp = 0.0
	if (Float64(t_1 + Float64(x - Float64(y / Float64(3.0 * z)))) <= -5e+269)
		tmp = Float64(x - Float64(Float64(Float64(y - Float64(t / y)) / z) / 3.0));
	else
		tmp = Float64(Float64(x - Float64(Float64(0.3333333333333333 * y) / z)) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = t / ((3.0 * z) * y);
	tmp = 0.0;
	if ((t_1 + (x - (y / (3.0 * z)))) <= -5e+269)
		tmp = x - (((y - (t / y)) / z) / 3.0);
	else
		tmp = (x - ((0.3333333333333333 * y) / z)) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+269], N[(x - N[(N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(N[(0.3333333333333333 * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + t$95$1), $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))) < -5.0000000000000002e269

   1. Initial program 89.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 89.2

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

   4. Applied rewrites 89.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. frac-2neg N/A

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

      14. lift-/.f64 N/A

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

      15. associate-+r- N/A

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

      16. lift--.f64 N/A

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

      17. +-commutative N/A

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

      18. lift--.f64 N/A

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

      19. associate-+l- N/A

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

      20. lower--.f64 N/A

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

   6. Applied rewrites 99.9%

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

   if -5.0000000000000002e269 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 98.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. div-inv N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 98.0

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

   4. Applied rewrites 98.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right) \leq -5 \cdot 10^{+269}:\\ \;\;\;\;x - \frac{\frac{y - \frac{t}{y}}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{0.3333333333333333 \cdot y}{z}\right) + \frac{t}{\left(3 \cdot z\right) \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+269}:\\ \;\;\;\;x - \frac{\frac{y - \frac{t}{y}}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ t (* (* 3.0 z) y)) (- x (/ y (* 3.0 z))))))
   (if (<= t_1 -5e+269) (- x (/ (/ (- y (/ t y)) z) 3.0)) t_1)))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t / ((3.0 * z) * y)) + (x - (y / (3.0 * z)));
	double tmp;
	if (t_1 <= -5e+269) {
		tmp = x - (((y - (t / y)) / z) / 3.0);
	} else {
		tmp = t_1;
	}
	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 = (t / ((3.0d0 * z) * y)) + (x - (y / (3.0d0 * z)))
    if (t_1 <= (-5d+269)) then
        tmp = x - (((y - (t / y)) / z) / 3.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t / ((3.0 * z) * y)) + (x - (y / (3.0 * z)));
	double tmp;
	if (t_1 <= -5e+269) {
		tmp = x - (((y - (t / y)) / z) / 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t / ((3.0 * z) * y)) + (x - (y / (3.0 * z)))
	tmp = 0
	if t_1 <= -5e+269:
		tmp = x - (((y - (t / y)) / z) / 3.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t / Float64(Float64(3.0 * z) * y)) + Float64(x - Float64(y / Float64(3.0 * z))))
	tmp = 0.0
	if (t_1 <= -5e+269)
		tmp = Float64(x - Float64(Float64(Float64(y - Float64(t / y)) / z) / 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t / ((3.0 * z) * y)) + (x - (y / (3.0 * z)));
	tmp = 0.0;
	if (t_1 <= -5e+269)
		tmp = x - (((y - (t / y)) / z) / 3.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+269], N[(x - N[(N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], t$95$1]]

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))) < -5.0000000000000002e269

   1. Initial program 89.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 89.2

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

   4. Applied rewrites 89.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. frac-2neg N/A

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

      14. lift-/.f64 N/A

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

      15. associate-+r- N/A

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

      16. lift--.f64 N/A

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

      17. +-commutative N/A

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

      18. lift--.f64 N/A

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

      19. associate-+l- N/A

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

      20. lower--.f64 N/A

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

   6. Applied rewrites 99.9%

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

   if -5.0000000000000002e269 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 98.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right) \leq -5 \cdot 10^{+269}:\\ \;\;\;\;x - \frac{\frac{y - \frac{t}{y}}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\left(3 \cdot z\right) \cdot y}\\ \mathbf{if}\;t\_1 + \left(x - \frac{y}{3 \cdot z}\right) \leq -5 \cdot 10^{+269}:\\ \;\;\;\;x - \frac{\frac{y - \frac{t}{y}}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, t\_1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ t (* (* 3.0 z) y))))
   (if (<= (+ t_1 (- x (/ y (* 3.0 z)))) -5e+269)
     (- x (/ (/ (- y (/ t y)) z) 3.0))
     (fma (/ -0.3333333333333333 z) y (+ t_1 x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = t / ((3.0 * z) * y);
	double tmp;
	if ((t_1 + (x - (y / (3.0 * z)))) <= -5e+269) {
		tmp = x - (((y - (t / y)) / z) / 3.0);
	} else {
		tmp = fma((-0.3333333333333333 / z), y, (t_1 + x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(t / Float64(Float64(3.0 * z) * y))
	tmp = 0.0
	if (Float64(t_1 + Float64(x - Float64(y / Float64(3.0 * z)))) <= -5e+269)
		tmp = Float64(x - Float64(Float64(Float64(y - Float64(t / y)) / z) / 3.0));
	else
		tmp = fma(Float64(-0.3333333333333333 / z), y, Float64(t_1 + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e+269], N[(x - N[(N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + N[(t$95$1 + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y))) < -5.0000000000000002e269

   1. Initial program 89.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 89.2

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

   4. Applied rewrites 89.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. frac-2neg N/A

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

      14. lift-/.f64 N/A

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

      15. associate-+r- N/A

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

      16. lift--.f64 N/A

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

      17. +-commutative N/A

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

      18. lift--.f64 N/A

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

      19. associate-+l- N/A

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

      20. lower--.f64 N/A

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

   6. Applied rewrites 99.9%

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

   if -5.0000000000000002e269 < (+.f64 (-.f64 x (/.f64 y (*.f64 z #s(literal 3 binary64)))) (/.f64 t (*.f64 (*.f64 z #s(literal 3 binary64)) y)))

   1. Initial program 98.0%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. distribute-lft-neg-in N/A

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

      10. distribute-frac-neg2 N/A

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

      11. lower-fma.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/r* N/A

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

      16. distribute-neg-frac N/A

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

      17. lower-/.f64 N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-+.f64 98.0

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 98.0

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

   4. Applied rewrites 98.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right) \leq -5 \cdot 10^{+269}:\\ \;\;\;\;x - \frac{\frac{y - \frac{t}{y}}{z}}{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, \frac{t}{\left(3 \cdot z\right) \cdot y} + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.3%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 88.4

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

   4. Applied rewrites 88.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. frac-2neg N/A

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

      14. lift-/.f64 N/A

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

      15. associate-+r- N/A

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

      16. lift--.f64 N/A

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

      17. +-commutative N/A

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

      18. lift--.f64 N/A

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

      19. associate-+l- N/A

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

      20. lower--.f64 N/A

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

   6. Applied rewrites 99.9%

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

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

   1. Initial program 98.0%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. +-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 97.9

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

      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. distribute-neg-frac N/A

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

      19. neg-mul-1 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. times-frac N/A

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   4. Applied rewrites 98.0%

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

4. Final simplification 98.3%

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

Alternative 6: 96.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.3%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 88.4

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

   4. Applied rewrites 88.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. frac-2neg N/A

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

      14. lift-/.f64 N/A

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

      15. associate-+r- N/A

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

      16. lift--.f64 N/A

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

      17. +-commutative N/A

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

      18. lift--.f64 N/A

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

      19. associate-+l- N/A

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

      20. lower--.f64 N/A

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

   6. Applied rewrites 99.9%

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

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

   1. Initial program 98.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 98.0

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

   4. Applied rewrites 98.0%

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

   6. Applied rewrites 97.5%

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

4. Final simplification 97.9%

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

Alternative 7: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{t}{y}}{z}\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, -0.3333333333333333, x\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{-t}{z}}{y} \cdot -0.3333333333333333 + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t\_1}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- y (/ t y)) z)))
   (if (<= y -1.7e-80)
     (fma t_1 -0.3333333333333333 x)
     (if (<= y 1.7e-73)
       (+ (* (/ (/ (- t) z) y) -0.3333333333333333) x)
       (- x (/ t_1 3.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - (t / y)) / z;
	double tmp;
	if (y <= -1.7e-80) {
		tmp = fma(t_1, -0.3333333333333333, x);
	} else if (y <= 1.7e-73) {
		tmp = (((-t / z) / y) * -0.3333333333333333) + x;
	} else {
		tmp = x - (t_1 / 3.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - Float64(t / y)) / z)
	tmp = 0.0
	if (y <= -1.7e-80)
		tmp = fma(t_1, -0.3333333333333333, x);
	elseif (y <= 1.7e-73)
		tmp = Float64(Float64(Float64(Float64(Float64(-t) / z) / y) * -0.3333333333333333) + x);
	else
		tmp = Float64(x - Float64(t_1 / 3.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, -1.7e-80], N[(t$95$1 * -0.3333333333333333 + x), $MachinePrecision], If[LessEqual[y, 1.7e-73], N[(N[(N[(N[((-t) / z), $MachinePrecision] / y), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(t$95$1 / 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7e-80

   1. Initial program 99.7%

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

   3. Taylor expanded in 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. distribute-lft-out-- N/A

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

      4. associate-/r* N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

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

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

      10. associate-*r/ N/A

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

   5. Applied rewrites 99.8%

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

   if -1.7e-80 < y < 1.7000000000000001e-73

   1. Initial program 93.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. distribute-lft-out-- N/A

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

      4. associate-/r* N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

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

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

      10. associate-*r/ N/A

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{t}{y \cdot z}, \frac{-1}{3}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 93.4%

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

   10. Applied rewrites 96.9%

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

   if 1.7000000000000001e-73 < y

   1. Initial program 97.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. metadata-eval 97.6

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

   4. Applied rewrites 97.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. frac-2neg N/A

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

      14. lift-/.f64 N/A

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

      15. associate-+r- N/A

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

      16. lift--.f64 N/A

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

      17. +-commutative N/A

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

      18. lift--.f64 N/A

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

      19. associate-+l- N/A

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

      20. lower--.f64 N/A

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

   6. Applied rewrites 98.8%

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

Alternative 8: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y - \frac{t}{y}\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_1}{z}, -0.3333333333333333, x\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{-t}{z}}{y} \cdot -0.3333333333333333 + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t\_1}{3 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- y (/ t y))))
   (if (<= y -1.7e-80)
     (fma (/ t_1 z) -0.3333333333333333 x)
     (if (<= y 1.8e-73)
       (+ (* (/ (/ (- t) z) y) -0.3333333333333333) x)
       (- x (/ t_1 (* 3.0 z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y - (t / y);
	double tmp;
	if (y <= -1.7e-80) {
		tmp = fma((t_1 / z), -0.3333333333333333, x);
	} else if (y <= 1.8e-73) {
		tmp = (((-t / z) / y) * -0.3333333333333333) + x;
	} else {
		tmp = x - (t_1 / (3.0 * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y - Float64(t / y))
	tmp = 0.0
	if (y <= -1.7e-80)
		tmp = fma(Float64(t_1 / z), -0.3333333333333333, x);
	elseif (y <= 1.8e-73)
		tmp = Float64(Float64(Float64(Float64(Float64(-t) / z) / y) * -0.3333333333333333) + x);
	else
		tmp = Float64(x - Float64(t_1 / Float64(3.0 * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.7e-80], N[(N[(t$95$1 / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], If[LessEqual[y, 1.8e-73], N[(N[(N[(N[((-t) / z), $MachinePrecision] / y), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(t$95$1 / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7e-80

   1. Initial program 99.7%

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

   3. Taylor expanded in 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. distribute-lft-out-- N/A

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

      4. associate-/r* N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

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

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

      10. associate-*r/ N/A

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

   5. Applied rewrites 99.8%

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

   if -1.7e-80 < y < 1.8e-73

   1. Initial program 93.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. distribute-lft-out-- N/A

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

      4. associate-/r* N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

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

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

      10. associate-*r/ N/A

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{t}{y \cdot z}, \frac{-1}{3}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 93.4%

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

   10. Applied rewrites 96.9%

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

   if 1.8e-73 < y

   1. Initial program 97.6%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 98.7

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

Alternative 9: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - \frac{t}{y}}{z}, -0.3333333333333333, x\right)\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{-80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{-t}{z}}{y} \cdot -0.3333333333333333 + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ (- y (/ t y)) z) -0.3333333333333333 x)))
   (if (<= y -1.7e-80)
     t_1
     (if (<= y 1.45e-62)
       (+ (* (/ (/ (- t) z) y) -0.3333333333333333) x)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(((y - (t / y)) / z), -0.3333333333333333, x);
	double tmp;
	if (y <= -1.7e-80) {
		tmp = t_1;
	} else if (y <= 1.45e-62) {
		tmp = (((-t / z) / y) * -0.3333333333333333) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(Float64(y - Float64(t / y)) / z), -0.3333333333333333, x)
	tmp = 0.0
	if (y <= -1.7e-80)
		tmp = t_1;
	elseif (y <= 1.45e-62)
		tmp = Float64(Float64(Float64(Float64(Float64(-t) / z) / y) * -0.3333333333333333) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]}, If[LessEqual[y, -1.7e-80], t$95$1, If[LessEqual[y, 1.45e-62], N[(N[(N[(N[((-t) / z), $MachinePrecision] / y), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e-80 or 1.44999999999999993e-62 < y

   1. Initial program 98.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. distribute-lft-out-- N/A

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

      4. associate-/r* N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

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

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

      10. associate-*r/ N/A

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

   5. Applied rewrites 99.1%

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

   if -1.7e-80 < y < 1.44999999999999993e-62

   1. Initial program 93.5%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. distribute-lft-out-- N/A

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

      4. associate-/r* N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

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

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

      10. associate-*r/ N/A

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

   5. Applied rewrites 87.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{t}{y \cdot z}, \frac{-1}{3}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 93.4%

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

   10. Applied rewrites 96.9%

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

Alternative 10: 89.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{3}, \frac{-1}{z}, x\right)\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z \cdot y}, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ y 3.0) (/ -1.0 z) x)))
   (if (<= y -2.5e-19)
     t_1
     (if (<= y 2.05e-14) (fma (/ (- t) (* z y)) -0.3333333333333333 x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y / 3.0), (-1.0 / z), x);
	double tmp;
	if (y <= -2.5e-19) {
		tmp = t_1;
	} else if (y <= 2.05e-14) {
		tmp = fma((-t / (z * y)), -0.3333333333333333, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y / 3.0), Float64(-1.0 / z), x)
	tmp = 0.0
	if (y <= -2.5e-19)
		tmp = t_1;
	elseif (y <= 2.05e-14)
		tmp = fma(Float64(Float64(-t) / Float64(z * y)), -0.3333333333333333, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / 3.0), $MachinePrecision] * N[(-1.0 / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -2.5e-19], t$95$1, If[LessEqual[y, 2.05e-14], N[(N[((-t) / N[(z * y), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5000000000000002e-19 or 2.0500000000000001e-14 < y

   1. Initial program 98.1%

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

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 94.1

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

   5. Applied rewrites 94.1%

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

   7. Applied rewrites 94.1%

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

   9. Applied rewrites 94.2%

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

   if -2.5000000000000002e-19 < y < 2.0500000000000001e-14

   1. Initial program 94.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. distribute-lft-out-- N/A

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

      4. associate-/r* N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

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

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

      10. associate-*r/ N/A

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{t}{y \cdot z}, \frac{-1}{3}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 91.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{3}, \frac{-1}{z}, x\right)\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z \cdot y}, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{3}, \frac{-1}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z \cdot y}, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{3 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.5e-19)
   (fma -0.3333333333333333 (/ y z) x)
   (if (<= y 2.05e-14)
     (fma (/ (- t) (* z y)) -0.3333333333333333 x)
     (- x (/ y (* 3.0 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.5e-19) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else if (y <= 2.05e-14) {
		tmp = fma((-t / (z * y)), -0.3333333333333333, x);
	} else {
		tmp = x - (y / (3.0 * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.5e-19)
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	elseif (y <= 2.05e-14)
		tmp = fma(Float64(Float64(-t) / Float64(z * y)), -0.3333333333333333, x);
	else
		tmp = Float64(x - Float64(y / Float64(3.0 * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.5e-19], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 2.05e-14], N[(N[((-t) / N[(z * y), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5000000000000002e-19

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 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

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 96.3

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

   5. Applied rewrites 96.3%

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

   if -2.5000000000000002e-19 < y < 2.0500000000000001e-14

   1. Initial program 94.8%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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. distribute-lft-out-- N/A

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

      4. associate-/r* N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

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

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

      10. associate-*r/ N/A

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{t}{y \cdot z}, \frac{-1}{3}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 91.0%

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

   if 2.0500000000000001e-14 < y

   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. 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

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 92.6

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

   5. Applied rewrites 92.6%

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

   7. Applied rewrites 92.6%

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

   9. Applied rewrites 92.6%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{z \cdot y}, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{3 \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{-63}:\\ \;\;\;\;\frac{t}{\left(3 \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{3 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -7e-60)
   (fma -0.3333333333333333 (/ y z) x)
   (if (<= y 3.55e-63) (/ t (* (* 3.0 z) y)) (- x (/ y (* 3.0 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7e-60) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else if (y <= 3.55e-63) {
		tmp = t / ((3.0 * z) * y);
	} else {
		tmp = x - (y / (3.0 * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7e-60)
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	elseif (y <= 3.55e-63)
		tmp = Float64(t / Float64(Float64(3.0 * z) * y));
	else
		tmp = Float64(x - Float64(y / Float64(3.0 * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -7e-60], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 3.55e-63], N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.99999999999999952e-60

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 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

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if -6.99999999999999952e-60 < y < 3.5500000000000001e-63

   1. Initial program 94.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   7. Applied rewrites 67.6%

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

   if 3.5500000000000001e-63 < y

   1. Initial program 97.5%

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

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 90.4

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

   5. Applied rewrites 90.4%

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

   7. Applied rewrites 90.4%

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

   9. Applied rewrites 90.4%

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

Alternative 13: 77.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{-63}:\\ \;\;\;\;\frac{t}{\left(3 \cdot y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{3 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -7e-60)
   (fma -0.3333333333333333 (/ y z) x)
   (if (<= y 3.55e-63) (/ t (* (* 3.0 y) z)) (- x (/ y (* 3.0 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7e-60) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else if (y <= 3.55e-63) {
		tmp = t / ((3.0 * y) * z);
	} else {
		tmp = x - (y / (3.0 * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7e-60)
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	elseif (y <= 3.55e-63)
		tmp = Float64(t / Float64(Float64(3.0 * y) * z));
	else
		tmp = Float64(x - Float64(y / Float64(3.0 * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -7e-60], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 3.55e-63], N[(t / N[(N[(3.0 * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.99999999999999952e-60

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 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

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if -6.99999999999999952e-60 < y < 3.5500000000000001e-63

   1. Initial program 94.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   7. Applied rewrites 67.6%

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

   9. Applied rewrites 67.5%

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

   if 3.5500000000000001e-63 < y

   1. Initial program 97.5%

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

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 90.4

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

   5. Applied rewrites 90.4%

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

   7. Applied rewrites 90.4%

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

   9. Applied rewrites 90.4%

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

Alternative 14: 77.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{-63}:\\ \;\;\;\;\frac{t}{z \cdot y} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{3 \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -7e-60)
   (fma -0.3333333333333333 (/ y z) x)
   (if (<= y 3.55e-63)
     (* (/ t (* z y)) 0.3333333333333333)
     (- x (/ y (* 3.0 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7e-60) {
		tmp = fma(-0.3333333333333333, (y / z), x);
	} else if (y <= 3.55e-63) {
		tmp = (t / (z * y)) * 0.3333333333333333;
	} else {
		tmp = x - (y / (3.0 * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7e-60)
		tmp = fma(-0.3333333333333333, Float64(y / z), x);
	elseif (y <= 3.55e-63)
		tmp = Float64(Float64(t / Float64(z * y)) * 0.3333333333333333);
	else
		tmp = Float64(x - Float64(y / Float64(3.0 * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -7e-60], N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 3.55e-63], N[(N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.99999999999999952e-60

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 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

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if -6.99999999999999952e-60 < y < 3.5500000000000001e-63

   1. Initial program 94.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if 3.5500000000000001e-63 < y

   1. Initial program 97.5%

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

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 90.4

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

   5. Applied rewrites 90.4%

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

   7. Applied rewrites 90.4%

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

   9. Applied rewrites 90.4%

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

Alternative 15: 95.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.5%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
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. distribute-lft-out-- N/A

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

   4. associate-/r* N/A

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

   5. div-sub N/A

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

   6. associate-/l* N/A

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

   7. metadata-eval N/A

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

   8. associate-*r* N/A

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

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

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

   10. associate-*r/ N/A

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

5. Applied rewrites 94.3%

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

Alternative 16: 95.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.5%

   \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
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. distribute-lft-out-- N/A

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

   4. associate-/r* N/A

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

   5. div-sub N/A

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

   6. associate-/l* N/A

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

   7. metadata-eval N/A

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

   8. associate-*r* N/A

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

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

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

   10. associate-*r/ N/A

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

5. Applied rewrites 94.3%

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

7. Applied rewrites 94.2%

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

Alternative 17: 64.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

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

   19. lower-fma.f64 N/A

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

   20. lower-/.f64 64.3

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

5. Applied rewrites 64.3%

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

Alternative 18: 36.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

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) * (-0.3333333333333333d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.5%

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

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

   19. lower-fma.f64 N/A

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

   20. lower-/.f64 64.3

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

5. Applied rewrites 64.3%

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

7. Applied rewrites 64.3%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 35.1%

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

Developer Target 1: 96.0% 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 2024288 
(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))))