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

Percentage Accurate: 95.2% → 97.7%

Time: 9.9s

Alternatives: 12

Speedup: 1.4×

• 27.8% 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: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{-153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t}{z}, 0.3333333333333333, y \cdot x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (- y (/ t y)) (* 3.0 z)))))
   (if (<= y -7.5e-153)
     t_1
     (if (<= y 1.35e-145)
       (/ (fma (/ t z) 0.3333333333333333 (* y x)) y)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - ((y - (t / y)) / (3.0 * z));
	double tmp;
	if (y <= -7.5e-153) {
		tmp = t_1;
	} else if (y <= 1.35e-145) {
		tmp = fma((t / z), 0.3333333333333333, (y * x)) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)))
	tmp = 0.0
	if (y <= -7.5e-153)
		tmp = t_1;
	elseif (y <= 1.35e-145)
		tmp = Float64(fma(Float64(t / z), 0.3333333333333333, Float64(y * x)) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e-153], t$95$1, If[LessEqual[y, 1.35e-145], N[(N[(N[(t / z), $MachinePrecision] * 0.3333333333333333 + N[(y * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.5e-153 or 1.35e-145 < y

   1. Initial program 97.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.3

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

      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 99.3

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

   4. Applied rewrites 99.3%

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

   if -7.5e-153 < y < 1.35e-145

   1. Initial program 92.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 96.4

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

   5. Applied rewrites 96.4%

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

Alternative 2: 96.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

def code(x, y, z, t):
	t_1 = (t / ((3.0 * z) * y)) - ((y / (3.0 * z)) - x)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (y - (t / y)) / (-3.0 * z)
	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(Float64(y / Float64(3.0 * z)) - x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(y - Float64(t / y)) / Float64(-3.0 * z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t / ((3.0 * z) * y)) - ((y / (3.0 * z)) - x);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (y - (t / y)) / (-3.0 * z);
	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[(N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(-3.0 * z), $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))) < -inf.0

   1. Initial program 85.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 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%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

   9. Applied rewrites 100.0%

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

   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 98.1%

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

4. Final simplification 98.3%

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

Alternative 3: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y - \frac{t}{y}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{t}{z}, 0.3333333333333333, y \cdot x\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_1}{z}, -0.3333333333333333, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- y (/ t y))))
   (if (<= y -7.5e-153)
     (fma t_1 (/ -0.3333333333333333 z) x)
     (if (<= y 3e-112)
       (/ (fma (/ t z) 0.3333333333333333 (* y x)) y)
       (fma (/ t_1 z) -0.3333333333333333 x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y - (t / y);
	double tmp;
	if (y <= -7.5e-153) {
		tmp = fma(t_1, (-0.3333333333333333 / z), x);
	} else if (y <= 3e-112) {
		tmp = fma((t / z), 0.3333333333333333, (y * x)) / y;
	} else {
		tmp = fma((t_1 / z), -0.3333333333333333, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y - Float64(t / y))
	tmp = 0.0
	if (y <= -7.5e-153)
		tmp = fma(t_1, Float64(-0.3333333333333333 / z), x);
	elseif (y <= 3e-112)
		tmp = Float64(fma(Float64(t / z), 0.3333333333333333, Float64(y * x)) / y);
	else
		tmp = fma(Float64(t_1 / z), -0.3333333333333333, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e-153], N[(t$95$1 * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 3e-112], N[(N[(N[(t / z), $MachinePrecision] * 0.3333333333333333 + N[(y * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(t$95$1 / z), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.5e-153

   1. Initial program 96.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 96.1

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

   4. Applied rewrites 96.1%

      \[\leadsto \left(x - \color{blue}{\frac{y \cdot 0.3333333333333333}{z}}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. metadata-eval N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. div-inv N/A

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

      13. lift-*.f64 N/A

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

      14. associate-+r- N/A

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

      15. +-commutative N/A

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

      16. lift-/.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-/l/ N/A

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

      19. lift-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-*.f64 N/A

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

   6. Applied rewrites 98.8%

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

   if -7.5e-153 < y < 3.0000000000000001e-112

   1. Initial program 91.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 96.7

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

   5. Applied rewrites 96.7%

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

   if 3.0000000000000001e-112 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in t 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%

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

Alternative 4: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-46}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 3.7e-46)
   (- x (/ (- y (/ t y)) (* 3.0 z)))
   (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 <= 3.7e-46) {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	} 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 (t <= 3.7e-46)
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	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[t, 3.7e-46], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $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 t < 3.69999999999999983e-46

   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. 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 97.8

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

      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 97.8

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

   4. Applied rewrites 97.8%

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

   if 3.69999999999999983e-46 < t

   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. 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 99.7

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

      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 99.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{-46}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z \cdot y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma -0.3333333333333333 (/ y z) x)))
   (if (<= y -1.7e+22)
     t_1
     (if (<= y 1.65e-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(-0.3333333333333333, (y / z), x);
	double tmp;
	if (y <= -1.7e+22) {
		tmp = t_1;
	} else if (y <= 1.65e-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(-0.3333333333333333, Float64(y / z), x)
	tmp = 0.0
	if (y <= -1.7e+22)
		tmp = t_1;
	elseif (y <= 1.65e-14)
		tmp = fma(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[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -1.7e+22], t$95$1, If[LessEqual[y, 1.65e-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 < -1.7e22 or 1.6499999999999999e-14 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 94.6

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

   5. Applied rewrites 94.6%

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

   if -1.7e22 < y < 1.6499999999999999e-14

   1. Initial program 91.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 89.1

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

      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 89.1

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

   4. Applied rewrites 89.1%

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

   5. Taylor expanded in y around 0

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

      1. div-sub N/A

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

      2. associate-*l/ N/A

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

      3. remove-double-neg N/A

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

      4. distribute-lft-neg-out N/A

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

      5. mul-1-neg N/A

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

      6. associate-/l* N/A

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

      7. associate-/l/ N/A

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

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

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. *-inverses N/A

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

      15. *-rgt-identity N/A

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

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

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

      18. *-commutative N/A

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

      19. mul-1-neg N/A

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

   7. Applied rewrites 89.1%

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

Alternative 6: 76.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{-85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{t}{\left(3 \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma -0.3333333333333333 (/ y z) x)))
   (if (<= y -5.2e-85) t_1 (if (<= y 3.7e-51) (/ t (* (* 3.0 z) y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(-0.3333333333333333, (y / z), x);
	double tmp;
	if (y <= -5.2e-85) {
		tmp = t_1;
	} else if (y <= 3.7e-51) {
		tmp = t / ((3.0 * z) * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(-0.3333333333333333, Float64(y / z), x)
	tmp = 0.0
	if (y <= -5.2e-85)
		tmp = t_1;
	elseif (y <= 3.7e-51)
		tmp = Float64(t / Float64(Float64(3.0 * z) * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -5.2e-85], t$95$1, If[LessEqual[y, 3.7e-51], N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.20000000000000023e-85 or 3.69999999999999973e-51 < y

   1. Initial program 98.7%

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

   3. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if -5.20000000000000023e-85 < y < 3.69999999999999973e-51

   1. Initial program 91.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   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 70.5

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

   5. Applied rewrites 70.5%

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

   7. Applied rewrites 70.6%

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

Alternative 7: 76.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{-85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{t}{\left(3 \cdot y\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma -0.3333333333333333 (/ y z) x)))
   (if (<= y -5.2e-85) t_1 (if (<= y 3.7e-51) (/ t (* (* 3.0 y) z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(-0.3333333333333333, (y / z), x);
	double tmp;
	if (y <= -5.2e-85) {
		tmp = t_1;
	} else if (y <= 3.7e-51) {
		tmp = t / ((3.0 * y) * z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(-0.3333333333333333, Float64(y / z), x)
	tmp = 0.0
	if (y <= -5.2e-85)
		tmp = t_1;
	elseif (y <= 3.7e-51)
		tmp = Float64(t / Float64(Float64(3.0 * y) * z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -5.2e-85], t$95$1, If[LessEqual[y, 3.7e-51], N[(t / N[(N[(3.0 * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.20000000000000023e-85 or 3.69999999999999973e-51 < y

   1. Initial program 98.7%

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

   3. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if -5.20000000000000023e-85 < y < 3.69999999999999973e-51

   1. Initial program 91.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   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 70.5

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

   5. Applied rewrites 70.5%

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

   7. Applied rewrites 70.6%

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

   9. Applied rewrites 70.6%

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

Alternative 8: 76.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{-85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{0.3333333333333333}{z \cdot y} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma -0.3333333333333333 (/ y z) x)))
   (if (<= y -5.2e-85)
     t_1
     (if (<= y 3.7e-51) (* (/ 0.3333333333333333 (* z y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(-0.3333333333333333, (y / z), x);
	double tmp;
	if (y <= -5.2e-85) {
		tmp = t_1;
	} else if (y <= 3.7e-51) {
		tmp = (0.3333333333333333 / (z * y)) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(-0.3333333333333333, Float64(y / z), x)
	tmp = 0.0
	if (y <= -5.2e-85)
		tmp = t_1;
	elseif (y <= 3.7e-51)
		tmp = Float64(Float64(0.3333333333333333 / Float64(z * y)) * t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -5.2e-85], t$95$1, If[LessEqual[y, 3.7e-51], N[(N[(0.3333333333333333 / N[(z * y), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.20000000000000023e-85 or 3.69999999999999973e-51 < y

   1. Initial program 98.7%

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

   3. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if -5.20000000000000023e-85 < y < 3.69999999999999973e-51

   1. Initial program 91.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{t}{y \cdot z}} \]
   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 70.5

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

   5. Applied rewrites 70.5%

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

   7. Applied rewrites 70.5%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{0.3333333333333333}{z \cdot y} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.9% 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 96.2%

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

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\left(x + \frac{1}{3} \cdot \frac{t}{y \cdot z}\right) - \frac{1}{3} \cdot \frac{y}{z}} \]
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 95.0%

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

Alternative 10: 64.0% 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 96.2%

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

3. Taylor expanded in t around 0

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

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

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

   2. metadata-eval N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 66.8

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

5. Applied rewrites 66.8%

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

Alternative 11: 35.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 45.1

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

5. Applied rewrites 45.1%

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

7. Applied rewrites 45.2%

   \[\leadsto \frac{y}{\color{blue}{-3 \cdot z}} \]
8. Add Preprocessing

Alternative 12: 34.9% 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 96.2%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 45.1

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

5. Applied rewrites 45.1%

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

6. Final simplification 45.1%

   \[\leadsto \frac{y}{z} \cdot -0.3333333333333333 \]
7. Add Preprocessing

Developer Target 1: 96.1% 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 2024268 
(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))))