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

Percentage Accurate: 95.4% → 99.2%

Time: 9.1s

Alternatives: 18

Speedup: 1.0×

• 27.9% 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: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\left(3 \cdot z\right) \cdot y}\\ \mathbf{if}\;3 \cdot z \leq -1 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, t\_1 + x\right)\\ \mathbf{elif}\;3 \cdot z \leq 2 \cdot 10^{-111}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(x - \frac{y}{3 \cdot z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ t (* (* 3.0 z) y))))
   (if (<= (* 3.0 z) -1e+82)
     (fma (/ -0.3333333333333333 z) y (+ t_1 x))
     (if (<= (* 3.0 z) 2e-111)
       (- x (/ (- y (/ t y)) (* 3.0 z)))
       (+ t_1 (- x (/ y (* 3.0 z))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = t / ((3.0 * z) * y);
	double tmp;
	if ((3.0 * z) <= -1e+82) {
		tmp = fma((-0.3333333333333333 / z), y, (t_1 + x));
	} else if ((3.0 * z) <= 2e-111) {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	} else {
		tmp = t_1 + (x - (y / (3.0 * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(t / Float64(Float64(3.0 * z) * y))
	tmp = 0.0
	if (Float64(3.0 * z) <= -1e+82)
		tmp = fma(Float64(-0.3333333333333333 / z), y, Float64(t_1 + x));
	elseif (Float64(3.0 * z) <= 2e-111)
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	else
		tmp = Float64(t_1 + Float64(x - Float64(y / Float64(3.0 * z))));
	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[(3.0 * z), $MachinePrecision], -1e+82], N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + N[(t$95$1 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(3.0 * z), $MachinePrecision], 2e-111], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(x - N[(y / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z #s(literal 3 binary64)) < -9.9999999999999996e81

   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. 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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

   if -9.9999999999999996e81 < (*.f64 z #s(literal 3 binary64)) < 2.00000000000000018e-111

   1. Initial program 90.4%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   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 99.9

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   if 2.00000000000000018e-111 < (*.f64 z #s(literal 3 binary64))

   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. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;3 \cdot z \leq -1 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, \frac{t}{\left(3 \cdot z\right) \cdot y} + x\right)\\ \mathbf{elif}\;3 \cdot z \leq 2 \cdot 10^{-111}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\left(3 \cdot z\right) \cdot y} + \left(x - \frac{y}{3 \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{1}{y \cdot 3}, \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, \frac{t}{\left(3 \cdot z\right) \cdot y} + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.95e-159)
   (fma (/ t z) (/ 1.0 (* y 3.0)) (fma -0.3333333333333333 (/ y z) x))
   (fma (/ -0.3333333333333333 z) y (+ (/ t (* (* 3.0 z) y)) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.95e-159) {
		tmp = fma((t / z), (1.0 / (y * 3.0)), fma(-0.3333333333333333, (y / z), x));
	} else {
		tmp = fma((-0.3333333333333333 / z), y, ((t / ((3.0 * z) * y)) + x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.95e-159)
		tmp = fma(Float64(t / z), Float64(1.0 / Float64(y * 3.0)), fma(-0.3333333333333333, Float64(y / z), x));
	else
		tmp = fma(Float64(-0.3333333333333333 / z), y, Float64(Float64(t / Float64(Float64(3.0 * z) * y)) + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.95e-159], N[(N[(t / z), $MachinePrecision] * N[(1.0 / N[(y * 3.0), $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + N[(N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.94999999999999988e-159

   1. Initial program 94.2%

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 98.1

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. lift-/.f64 N/A

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

      17. distribute-neg-frac N/A

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

      18. neg-mul-1 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. times-frac N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

      24. metadata-eval N/A

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

      25. lower-fma.f64 N/A

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

   4. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{z}, \frac{1}{3 \cdot y}, \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\right)} \]

   if 1.94999999999999988e-159 < t

   1. Initial program 96.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. 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 99.1

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{1}{y \cdot 3}, \mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, \frac{t}{\left(3 \cdot z\right) \cdot y} + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, \frac{t}{\left(3 \cdot z\right) \cdot y} + x\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-112}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ -0.3333333333333333 z) y (+ (/ t (* (* 3.0 z) y)) x))))
   (if (<= t -1e+95)
     t_1
     (if (<= t 7.2e-112) (- x (/ (- y (/ t y)) (* 3.0 z))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((-0.3333333333333333 / z), y, ((t / ((3.0 * z) * y)) + x));
	double tmp;
	if (t <= -1e+95) {
		tmp = t_1;
	} else if (t <= 7.2e-112) {
		tmp = x - ((y - (t / y)) / (3.0 * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(-0.3333333333333333 / z), y, Float64(Float64(t / Float64(Float64(3.0 * z) * y)) + x))
	tmp = 0.0
	if (t <= -1e+95)
		tmp = t_1;
	elseif (t <= 7.2e-112)
		tmp = Float64(x - Float64(Float64(y - Float64(t / y)) / Float64(3.0 * z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + N[(N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1e+95], t$95$1, If[LessEqual[t, 7.2e-112], N[(x - N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / N[(3.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.00000000000000002e95 or 7.2000000000000002e-112 < t

   1. Initial program 98.2%

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

      1. lift-+.f64 N/A

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

      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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   if -1.00000000000000002e95 < t < 7.2000000000000002e-112

   1. Initial program 91.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, \frac{t}{\left(3 \cdot z\right) \cdot y} + x\right)\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-112}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, \frac{t}{\left(3 \cdot z\right) \cdot y} + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) - \frac{\frac{t}{-3 \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, \frac{t}{\left(3 \cdot z\right) \cdot y} + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.95e-159)
   (- (fma -0.3333333333333333 (/ y z) x) (/ (/ t (* -3.0 z)) y))
   (fma (/ -0.3333333333333333 z) y (+ (/ t (* (* 3.0 z) y)) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.95e-159) {
		tmp = fma(-0.3333333333333333, (y / z), x) - ((t / (-3.0 * z)) / y);
	} else {
		tmp = fma((-0.3333333333333333 / z), y, ((t / ((3.0 * z) * y)) + x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.95e-159)
		tmp = Float64(fma(-0.3333333333333333, Float64(y / z), x) - Float64(Float64(t / Float64(-3.0 * z)) / y));
	else
		tmp = fma(Float64(-0.3333333333333333 / z), y, Float64(Float64(t / Float64(Float64(3.0 * z) * y)) + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.95e-159], N[(N[(-0.3333333333333333 * N[(y / z), $MachinePrecision] + x), $MachinePrecision] - N[(N[(t / N[(-3.0 * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + N[(N[(t / N[(N[(3.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.94999999999999988e-159

   1. Initial program 94.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 98.0%

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

   if 1.94999999999999988e-159 < t

   1. Initial program 96.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. 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 99.1

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

      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 99.1

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

   4. Applied rewrites 99.1%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333, \frac{y}{z}, x\right) - \frac{\frac{t}{-3 \cdot z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, \frac{t}{\left(3 \cdot z\right) \cdot y} + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{-87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.85 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{t}{z}}{y \cdot 3} + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (- y (/ t y)) (* 3.0 z)))))
   (if (<= y -1.5e-87)
     t_1
     (if (<= y 3.85e-187) (+ (/ (/ t z) (* y 3.0)) x) 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 <= -1.5e-87) {
		tmp = t_1;
	} else if (y <= 3.85e-187) {
		tmp = ((t / z) / (y * 3.0)) + x;
	} 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 = x - ((y - (t / y)) / (3.0d0 * z))
    if (y <= (-1.5d-87)) then
        tmp = t_1
    else if (y <= 3.85d-187) then
        tmp = ((t / z) / (y * 3.0d0)) + x
    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 = x - ((y - (t / y)) / (3.0 * z));
	double tmp;
	if (y <= -1.5e-87) {
		tmp = t_1;
	} else if (y <= 3.85e-187) {
		tmp = ((t / z) / (y * 3.0)) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - ((y - (t / y)) / (3.0 * z))
	tmp = 0
	if y <= -1.5e-87:
		tmp = t_1
	elif y <= 3.85e-187:
		tmp = ((t / z) / (y * 3.0)) + x
	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 <= -1.5e-87)
		tmp = t_1;
	elseif (y <= 3.85e-187)
		tmp = Float64(Float64(Float64(t / z) / Float64(y * 3.0)) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - ((y - (t / y)) / (3.0 * z));
	tmp = 0.0;
	if (y <= -1.5e-87)
		tmp = t_1;
	elseif (y <= 3.85e-187)
		tmp = ((t / z) / (y * 3.0)) + x;
	else
		tmp = t_1;
	end
	tmp_2 = 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, -1.5e-87], t$95$1, If[LessEqual[y, 3.85e-187], N[(N[(N[(t / z), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.50000000000000008e-87 or 3.84999999999999982e-187 < y

   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. 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 -1.50000000000000008e-87 < y < 3.84999999999999982e-187

   1. Initial program 91.6%

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

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 98.4%

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

   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

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate-/l/ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 91.3

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

   7. Applied rewrites 91.3%

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

   9. Applied rewrites 91.6%

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

   11. Applied rewrites 98.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-87}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \mathbf{elif}\;y \leq 3.85 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{t}{z}}{y \cdot 3} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - \frac{t}{y}}{3 \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - \frac{t}{y}, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{if}\;y \leq -4.2 \cdot 10^{-84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.85 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{t}{z}}{y \cdot 3} + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (- y (/ t y)) (/ -0.3333333333333333 z) x)))
   (if (<= y -4.2e-84)
     t_1
     (if (<= y 3.85e-187) (+ (/ (/ t z) (* y 3.0)) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y - (t / y)), (-0.3333333333333333 / z), x);
	double tmp;
	if (y <= -4.2e-84) {
		tmp = t_1;
	} else if (y <= 3.85e-187) {
		tmp = ((t / z) / (y * 3.0)) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y - Float64(t / y)), Float64(-0.3333333333333333 / z), x)
	tmp = 0.0
	if (y <= -4.2e-84)
		tmp = t_1;
	elseif (y <= 3.85e-187)
		tmp = Float64(Float64(Float64(t / z) / Float64(y * 3.0)) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[y, -4.2e-84], t$95$1, If[LessEqual[y, 3.85e-187], N[(N[(N[(t / z), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.19999999999999996e-84 or 3.84999999999999982e-187 < y

   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 99.2%

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

   7. Applied rewrites 99.2%

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

   if -4.19999999999999996e-84 < y < 3.84999999999999982e-187

   1. Initial program 91.6%

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

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 98.4%

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

   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

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate-/l/ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 91.3

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

   7. Applied rewrites 91.3%

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

   9. Applied rewrites 91.6%

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

   11. Applied rewrites 98.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-84}:\\ \;\;\;\;\mathsf{fma}\left(y - \frac{t}{y}, \frac{-0.3333333333333333}{z}, x\right)\\ \mathbf{elif}\;y \leq 3.85 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{t}{z}}{y \cdot 3} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - \frac{t}{y}, \frac{-0.3333333333333333}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\frac{y}{3}}{z}\\ \mathbf{if}\;y \leq -3.45 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{t}{z}}{y \cdot 3} + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (/ y 3.0) z))))
   (if (<= y -3.45e+46)
     t_1
     (if (<= y 6.3e+44) (+ (/ (/ t z) (* y 3.0)) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - ((y / 3.0) / z);
	double tmp;
	if (y <= -3.45e+46) {
		tmp = t_1;
	} else if (y <= 6.3e+44) {
		tmp = ((t / z) / (y * 3.0)) + x;
	} 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 = x - ((y / 3.0d0) / z)
    if (y <= (-3.45d+46)) then
        tmp = t_1
    else if (y <= 6.3d+44) then
        tmp = ((t / z) / (y * 3.0d0)) + x
    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 = x - ((y / 3.0) / z);
	double tmp;
	if (y <= -3.45e+46) {
		tmp = t_1;
	} else if (y <= 6.3e+44) {
		tmp = ((t / z) / (y * 3.0)) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - ((y / 3.0) / z)
	tmp = 0
	if y <= -3.45e+46:
		tmp = t_1
	elif y <= 6.3e+44:
		tmp = ((t / z) / (y * 3.0)) + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(Float64(y / 3.0) / z))
	tmp = 0.0
	if (y <= -3.45e+46)
		tmp = t_1;
	elseif (y <= 6.3e+44)
		tmp = Float64(Float64(Float64(t / z) / Float64(y * 3.0)) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - ((y / 3.0) / z);
	tmp = 0.0;
	if (y <= -3.45e+46)
		tmp = t_1;
	elseif (y <= 6.3e+44)
		tmp = ((t / z) / (y * 3.0)) + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[(y / 3.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.45e+46], t$95$1, If[LessEqual[y, 6.3e+44], N[(N[(N[(t / z), $MachinePrecision] / N[(y * 3.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.45000000000000009e46 or 6.3e44 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 97.2

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

   7. Applied rewrites 97.2%

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

   9. Applied rewrites 97.4%

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

   if -3.45000000000000009e46 < y < 6.3e44

   1. Initial program 91.6%

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

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 96.5%

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

   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

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate-/l/ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 87.5

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

   7. Applied rewrites 87.5%

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

   9. Applied rewrites 87.6%

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

   11. Applied rewrites 91.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.45 \cdot 10^{+46}:\\ \;\;\;\;x - \frac{\frac{y}{3}}{z}\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{t}{z}}{y \cdot 3} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{y}{3}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\frac{y}{3}}{z}\\ \mathbf{if}\;y \leq -3.45 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{t}{z}}{y}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (/ y 3.0) z))))
   (if (<= y -3.45e+46)
     t_1
     (if (<= y 6.3e+44) (fma (/ (/ t z) y) 0.3333333333333333 x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - ((y / 3.0) / z);
	double tmp;
	if (y <= -3.45e+46) {
		tmp = t_1;
	} else if (y <= 6.3e+44) {
		tmp = fma(((t / z) / y), 0.3333333333333333, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(Float64(y / 3.0) / z))
	tmp = 0.0
	if (y <= -3.45e+46)
		tmp = t_1;
	elseif (y <= 6.3e+44)
		tmp = fma(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[(x - N[(N[(y / 3.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.45e+46], t$95$1, If[LessEqual[y, 6.3e+44], N[(N[(N[(t / z), $MachinePrecision] / y), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.45000000000000009e46 or 6.3e44 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 97.2

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

   7. Applied rewrites 97.2%

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

   9. Applied rewrites 97.4%

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

   if -3.45000000000000009e46 < y < 6.3e44

   1. Initial program 91.6%

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

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 96.5%

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

   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

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate-/l/ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 87.5

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

   7. Applied rewrites 87.5%

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

   9. Applied rewrites 91.8%

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

Alternative 9: 91.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\frac{y}{3}}{z}\\ \mathbf{if}\;y \leq -3.45 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{0.3333333333333333}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (/ y 3.0) z))))
   (if (<= y -3.45e+46)
     t_1
     (if (<= y 6.3e+44) (fma (/ t z) (/ 0.3333333333333333 y) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - ((y / 3.0) / z);
	double tmp;
	if (y <= -3.45e+46) {
		tmp = t_1;
	} else if (y <= 6.3e+44) {
		tmp = fma((t / z), (0.3333333333333333 / y), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(Float64(y / 3.0) / z))
	tmp = 0.0
	if (y <= -3.45e+46)
		tmp = t_1;
	elseif (y <= 6.3e+44)
		tmp = fma(Float64(t / z), Float64(0.3333333333333333 / y), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[(y / 3.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.45e+46], t$95$1, If[LessEqual[y, 6.3e+44], N[(N[(t / z), $MachinePrecision] * N[(0.3333333333333333 / y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.45000000000000009e46 or 6.3e44 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 97.2

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

   7. Applied rewrites 97.2%

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

   9. Applied rewrites 97.4%

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

   if -3.45000000000000009e46 < y < 6.3e44

   1. Initial program 91.6%

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

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 96.5%

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

   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

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate-/l/ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 87.5

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

   7. Applied rewrites 87.5%

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

   9. Applied rewrites 91.7%

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

Alternative 10: 89.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\frac{y}{3}}{z}\\ \mathbf{if}\;y \leq -3.45 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{t}{\left(y \cdot 3\right) \cdot z} + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (/ y 3.0) z))))
   (if (<= y -3.45e+46)
     t_1
     (if (<= y 6.6e+44) (+ (/ t (* (* y 3.0) z)) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - ((y / 3.0) / z);
	double tmp;
	if (y <= -3.45e+46) {
		tmp = t_1;
	} else if (y <= 6.6e+44) {
		tmp = (t / ((y * 3.0) * z)) + x;
	} 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 = x - ((y / 3.0d0) / z)
    if (y <= (-3.45d+46)) then
        tmp = t_1
    else if (y <= 6.6d+44) then
        tmp = (t / ((y * 3.0d0) * z)) + x
    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 = x - ((y / 3.0) / z);
	double tmp;
	if (y <= -3.45e+46) {
		tmp = t_1;
	} else if (y <= 6.6e+44) {
		tmp = (t / ((y * 3.0) * z)) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - ((y / 3.0) / z)
	tmp = 0
	if y <= -3.45e+46:
		tmp = t_1
	elif y <= 6.6e+44:
		tmp = (t / ((y * 3.0) * z)) + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(Float64(y / 3.0) / z))
	tmp = 0.0
	if (y <= -3.45e+46)
		tmp = t_1;
	elseif (y <= 6.6e+44)
		tmp = Float64(Float64(t / Float64(Float64(y * 3.0) * z)) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - ((y / 3.0) / z);
	tmp = 0.0;
	if (y <= -3.45e+46)
		tmp = t_1;
	elseif (y <= 6.6e+44)
		tmp = (t / ((y * 3.0) * z)) + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[(y / 3.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.45e+46], t$95$1, If[LessEqual[y, 6.6e+44], N[(N[(t / N[(N[(y * 3.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.45000000000000009e46 or 6.60000000000000027e44 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 97.2

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

   7. Applied rewrites 97.2%

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

   9. Applied rewrites 97.4%

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

   if -3.45000000000000009e46 < y < 6.60000000000000027e44

   1. Initial program 91.6%

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

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 96.5%

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

   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

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate-/l/ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 87.5

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

   7. Applied rewrites 87.5%

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

   9. Applied rewrites 87.6%

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

   11. Applied rewrites 87.6%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.45 \cdot 10^{+46}:\\ \;\;\;\;x - \frac{\frac{y}{3}}{z}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{t}{\left(y \cdot 3\right) \cdot z} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{y}{3}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.45 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, x\right)\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{t}{\left(y \cdot 3\right) \cdot z} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{0.3333333333333333 \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.45e+46)
   (fma (/ -0.3333333333333333 z) y x)
   (if (<= y 6.6e+44)
     (+ (/ t (* (* y 3.0) z)) x)
     (- x (/ (* 0.3333333333333333 y) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.45e+46) {
		tmp = fma((-0.3333333333333333 / z), y, x);
	} else if (y <= 6.6e+44) {
		tmp = (t / ((y * 3.0) * z)) + x;
	} else {
		tmp = x - ((0.3333333333333333 * y) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.45e+46)
		tmp = fma(Float64(-0.3333333333333333 / z), y, x);
	elseif (y <= 6.6e+44)
		tmp = Float64(Float64(t / Float64(Float64(y * 3.0) * z)) + x);
	else
		tmp = Float64(x - Float64(Float64(0.3333333333333333 * y) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.45e+46], N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[y, 6.6e+44], N[(N[(t / N[(N[(y * 3.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(N[(0.3333333333333333 * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.45000000000000009e46

   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 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 96.8

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

   5. Applied rewrites 96.8%

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

   7. Applied rewrites 96.9%

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

   if -3.45000000000000009e46 < y < 6.60000000000000027e44

   1. Initial program 91.6%

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

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 96.5%

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

   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

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate-/l/ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 87.5

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

   7. Applied rewrites 87.5%

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

   9. Applied rewrites 87.6%

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

   11. Applied rewrites 87.6%

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

   if 6.60000000000000027e44 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 97.7

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

   7. Applied rewrites 97.7%

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

   9. Applied rewrites 97.8%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.45 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, x\right)\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{t}{\left(y \cdot 3\right) \cdot z} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{0.3333333333333333 \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.45 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, x\right)\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y \cdot z}, 0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{0.3333333333333333 \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.45e+46)
   (fma (/ -0.3333333333333333 z) y x)
   (if (<= y 6.6e+44)
     (fma (/ t (* y z)) 0.3333333333333333 x)
     (- x (/ (* 0.3333333333333333 y) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.45e+46) {
		tmp = fma((-0.3333333333333333 / z), y, x);
	} else if (y <= 6.6e+44) {
		tmp = fma((t / (y * z)), 0.3333333333333333, x);
	} else {
		tmp = x - ((0.3333333333333333 * y) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.45e+46)
		tmp = fma(Float64(-0.3333333333333333 / z), y, x);
	elseif (y <= 6.6e+44)
		tmp = fma(Float64(t / Float64(y * z)), 0.3333333333333333, x);
	else
		tmp = Float64(x - Float64(Float64(0.3333333333333333 * y) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.45e+46], N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[y, 6.6e+44], N[(N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision], N[(x - N[(N[(0.3333333333333333 * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.45000000000000009e46

   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 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 96.8

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

   5. Applied rewrites 96.8%

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

   7. Applied rewrites 96.9%

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

   if -3.45000000000000009e46 < y < 6.60000000000000027e44

   1. Initial program 91.6%

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

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 96.5%

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

   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

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate-/l/ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 87.5

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

   7. Applied rewrites 87.5%

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

   if 6.60000000000000027e44 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 97.7

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

   7. Applied rewrites 97.7%

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

   9. Applied rewrites 97.8%

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

Alternative 13: 88.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.45 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, x\right)\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{y \cdot z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{0.3333333333333333 \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.45e+46)
   (fma (/ -0.3333333333333333 z) y x)
   (if (<= y 6.6e+44)
     (fma t (/ 0.3333333333333333 (* y z)) x)
     (- x (/ (* 0.3333333333333333 y) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.45e+46) {
		tmp = fma((-0.3333333333333333 / z), y, x);
	} else if (y <= 6.6e+44) {
		tmp = fma(t, (0.3333333333333333 / (y * z)), x);
	} else {
		tmp = x - ((0.3333333333333333 * y) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.45e+46)
		tmp = fma(Float64(-0.3333333333333333 / z), y, x);
	elseif (y <= 6.6e+44)
		tmp = fma(t, Float64(0.3333333333333333 / Float64(y * z)), x);
	else
		tmp = Float64(x - Float64(Float64(0.3333333333333333 * y) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.45e+46], N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[y, 6.6e+44], N[(t * N[(0.3333333333333333 / N[(y * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(N[(0.3333333333333333 * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.45000000000000009e46

   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 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 96.8

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

   5. Applied rewrites 96.8%

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

   7. Applied rewrites 96.9%

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

   if -3.45000000000000009e46 < y < 6.60000000000000027e44

   1. Initial program 91.6%

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

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

      5. associate--r+ N/A

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

      6. lift--.f64 N/A

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

      7. lower--.f64 N/A

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

   4. Applied rewrites 96.5%

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

   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

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

      3. *-inverses N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/l* N/A

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

      6. associate-/l/ N/A

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

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

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 87.5

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

   7. Applied rewrites 87.5%

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

   9. Applied rewrites 86.6%

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

   if 6.60000000000000027e44 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 97.7

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

   7. Applied rewrites 97.7%

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

   9. Applied rewrites 97.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.45 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, x\right)\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{0.3333333333333333}{y \cdot z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{0.3333333333333333 \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, x\right)\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-71}:\\ \;\;\;\;\frac{t}{y \cdot z} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;x - \frac{0.3333333333333333 \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.2e-96)
   (fma (/ -0.3333333333333333 z) y x)
   (if (<= y 2.45e-71)
     (* (/ t (* y z)) 0.3333333333333333)
     (- x (/ (* 0.3333333333333333 y) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.2e-96) {
		tmp = fma((-0.3333333333333333 / z), y, x);
	} else if (y <= 2.45e-71) {
		tmp = (t / (y * z)) * 0.3333333333333333;
	} else {
		tmp = x - ((0.3333333333333333 * y) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.2e-96)
		tmp = fma(Float64(-0.3333333333333333 / z), y, x);
	elseif (y <= 2.45e-71)
		tmp = Float64(Float64(t / Float64(y * z)) * 0.3333333333333333);
	else
		tmp = Float64(x - Float64(Float64(0.3333333333333333 * y) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.2e-96], N[(N[(-0.3333333333333333 / z), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[y, 2.45e-71], N[(N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(x - N[(N[(0.3333333333333333 * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.19999999999999979e-96

   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 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 87.6

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

   5. Applied rewrites 87.6%

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

   7. Applied rewrites 87.7%

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

   if -2.19999999999999979e-96 < y < 2.4499999999999999e-71

   1. Initial program 89.5%

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

   3. Taylor expanded in y around 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 74.3

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

   5. Applied rewrites 74.3%

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

   if 2.4499999999999999e-71 < y

   1. Initial program 98.3%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 92.2

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

   7. Applied rewrites 92.2%

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

   9. Applied rewrites 92.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-96}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-0.3333333333333333}{z}, y, x\right)\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-71}:\\ \;\;\;\;\frac{t}{y \cdot z} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;x - \frac{0.3333333333333333 \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.1% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
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 63.1

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

5. Applied rewrites 63.1%

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

7. Applied rewrites 63.1%

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

Alternative 16: 64.1% 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 95.2%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
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 63.1

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

5. Applied rewrites 63.1%

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

Alternative 17: 35.9% 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 95.2%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
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 63.1

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

5. Applied rewrites 63.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 36.9%

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

10. Applied rewrites 37.0%

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

Alternative 18: 35.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (-0.3333333333333333 / z) * 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 = ((-0.3333333333333333d0) / z) * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{y \cdot \left(\frac{x}{y} - \frac{1}{3} \cdot \frac{1}{z}\right)} \]
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 63.1

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

5. Applied rewrites 63.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 36.9%

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

10. Applied rewrites 36.9%

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

11. Final simplification 36.9%

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

Developer Target 1: 96.2% 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 2024296 
(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))))