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

Percentage Accurate: 91.9% → 97.5%

Time: 10.9s

Alternatives: 12

Speedup: 0.7×

• 28.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) - ((z * 9.0d0) * t)) / (a * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Python

def code(x, y, z, t, a):
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0)

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((x * y) - ((z * 9.0d0) * t)) / (a * 2.0d0)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

Python

def code(x, y, z, t, a):
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0)

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ t_2 := x \cdot \frac{y}{a \cdot 2}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{z \cdot 4.5}{a}, t\_2\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, z \cdot \left(-4.5\right), t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))) (t_2 (* x (/ y (* a 2.0)))))
   (if (<= t_1 (- INFINITY))
     (fma (- t) (/ (* z 4.5) a) t_2)
     (if (<= t_1 2e+297)
       (/ (fma (* z -9.0) t (* x y)) (* a 2.0))
       (fma (/ t a) (* z (- 4.5)) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * y) - ((z * 9.0) * t);
	double t_2 = x * (y / (a * 2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(-t, ((z * 4.5) / a), t_2);
	} else if (t_1 <= 2e+297) {
		tmp = fma((z * -9.0), t, (x * y)) / (a * 2.0);
	} else {
		tmp = fma((t / a), (z * -4.5), t_2);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t))
	t_2 = Float64(x * Float64(y / Float64(a * 2.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(-t), Float64(Float64(z * 4.5) / a), t_2);
	elseif (t_1 <= 2e+297)
		tmp = Float64(fma(Float64(z * -9.0), t, Float64(x * y)) / Float64(a * 2.0));
	else
		tmp = fma(Float64(t / a), Float64(z * Float64(-4.5)), t_2);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(y / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[((-t) * N[(N[(z * 4.5), $MachinePrecision] / a), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$1, 2e+297], N[(N[(N[(z * -9.0), $MachinePrecision] * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(t / a), $MachinePrecision] * N[(z * (-4.5)), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -inf.0

   1. Initial program 66.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{\color{blue}{a \cdot 2}} \]

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} + \left(\mathsf{neg}\left(\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right)\right) + \frac{x \cdot y}{a \cdot 2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2}\right)\right) + \frac{x \cdot y}{a \cdot 2} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{t \cdot \left(z \cdot 9\right)}}{a \cdot 2}\right)\right) + \frac{x \cdot y}{a \cdot 2} \]

      10. associate-/l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{t \cdot \frac{z \cdot 9}{a \cdot 2}}\right)\right) + \frac{x \cdot y}{a \cdot 2} \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t\right)\right) \cdot \frac{z \cdot 9}{a \cdot 2}} + \frac{x \cdot y}{a \cdot 2} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(t\right), \frac{z \cdot 9}{a \cdot 2}, \frac{x \cdot y}{a \cdot 2}\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-t, \frac{z \cdot 4.5}{a}, x \cdot \frac{y}{a \cdot 2}\right)} \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < 2e297

   1. Initial program 98.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. sub-neg N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right) + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 9\right) \cdot t}\right)\right) + x \cdot y}{a \cdot 2} \]

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

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t} + x \cdot y}{a \cdot 2} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z \cdot 9\right), t, x \cdot y\right)}}{a \cdot 2} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right), t, x \cdot y\right)}{a \cdot 2} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(9\right)\right)}, t, x \cdot y\right)}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(9\right)\right)}, t, x \cdot y\right)}{a \cdot 2} \]

      12. metadata-eval 98.3

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot \color{blue}{-9}, t, x \cdot y\right)}{a \cdot 2} \]

   4. Applied rewrites 98.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}}{a \cdot 2} \]

   if 2e297 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 73.4%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{\color{blue}{a \cdot 2}} \]

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} + \left(\mathsf{neg}\left(\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right)\right) + \frac{x \cdot y}{a \cdot 2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2}\right)\right) + \frac{x \cdot y}{a \cdot 2} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{t \cdot \left(z \cdot 9\right)}}{a \cdot 2}\right)\right) + \frac{x \cdot y}{a \cdot 2} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{t \cdot \left(z \cdot 9\right)}{\color{blue}{a \cdot 2}}\right)\right) + \frac{x \cdot y}{a \cdot 2} \]

      11. times-frac N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{t}{a} \cdot \frac{z \cdot 9}{2}}\right)\right) + \frac{x \cdot y}{a \cdot 2} \]

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

          \[\leadsto \color{blue}{\frac{t}{a} \cdot \left(\mathsf{neg}\left(\frac{z \cdot 9}{2}\right)\right)} + \frac{x \cdot y}{a \cdot 2} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{a}, \mathsf{neg}\left(\frac{z \cdot 9}{2}\right), \frac{x \cdot y}{a \cdot 2}\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t}{a}}, \mathsf{neg}\left(\frac{z \cdot 9}{2}\right), \frac{x \cdot y}{a \cdot 2}\right) \]

      15. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{a}, \color{blue}{\mathsf{neg}\left(\frac{z \cdot 9}{2}\right)}, \frac{x \cdot y}{a \cdot 2}\right) \]

      16. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{a}, \mathsf{neg}\left(\frac{\color{blue}{z \cdot 9}}{2}\right), \frac{x \cdot y}{a \cdot 2}\right) \]

      17. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{a}, \mathsf{neg}\left(\color{blue}{z \cdot \frac{9}{2}}\right), \frac{x \cdot y}{a \cdot 2}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{a}, \mathsf{neg}\left(\color{blue}{z \cdot \frac{9}{2}}\right), \frac{x \cdot y}{a \cdot 2}\right) \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{a}, \mathsf{neg}\left(z \cdot \color{blue}{\frac{9}{2}}\right), \frac{x \cdot y}{a \cdot 2}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t}{a}, \mathsf{neg}\left(z \cdot \frac{9}{2}\right), \frac{\color{blue}{x \cdot y}}{a \cdot 2}\right) \]

   4. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{a}, -z \cdot 4.5, x \cdot \frac{y}{a \cdot 2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{z \cdot 4.5}{a}, x \cdot \frac{y}{a \cdot 2}\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2 \cdot 10^{+297}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, z \cdot \left(-4.5\right), x \cdot \frac{y}{a \cdot 2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{z \cdot 4.5}{a}, x \cdot \frac{y}{a \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- (* x y) (* (* z 9.0) t)) (- INFINITY))
   (fma (- t) (/ (* z 4.5) a) (* x (/ y (* a 2.0))))
   (/ (fma (* z -9.0) t (* x y)) (* a 2.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) - ((z * 9.0) * t)) <= -((double) INFINITY)) {
		tmp = fma(-t, ((z * 4.5) / a), (x * (y / (a * 2.0))));
	} else {
		tmp = fma((z * -9.0), t, (x * y)) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) <= Float64(-Inf))
		tmp = fma(Float64(-t), Float64(Float64(z * 4.5) / a), Float64(x * Float64(y / Float64(a * 2.0))));
	else
		tmp = Float64(fma(Float64(z * -9.0), t, Float64(x * y)) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[((-t) * N[(N[(z * 4.5), $MachinePrecision] / a), $MachinePrecision] + N[(x * N[(y / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * -9.0), $MachinePrecision] * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -inf.0

   1. Initial program 66.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{\color{blue}{a \cdot 2}} \]

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} + \left(\mathsf{neg}\left(\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right)\right) + \frac{x \cdot y}{a \cdot 2}} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2}\right)\right) + \frac{x \cdot y}{a \cdot 2} \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{t \cdot \left(z \cdot 9\right)}}{a \cdot 2}\right)\right) + \frac{x \cdot y}{a \cdot 2} \]

      10. associate-/l* N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{t \cdot \frac{z \cdot 9}{a \cdot 2}}\right)\right) + \frac{x \cdot y}{a \cdot 2} \]

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

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(t\right)\right) \cdot \frac{z \cdot 9}{a \cdot 2}} + \frac{x \cdot y}{a \cdot 2} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(t\right), \frac{z \cdot 9}{a \cdot 2}, \frac{x \cdot y}{a \cdot 2}\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-t, \frac{z \cdot 4.5}{a}, x \cdot \frac{y}{a \cdot 2}\right)} \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 95.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. sub-neg N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right) + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 9\right) \cdot t}\right)\right) + x \cdot y}{a \cdot 2} \]

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

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t} + x \cdot y}{a \cdot 2} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z \cdot 9\right), t, x \cdot y\right)}}{a \cdot 2} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right), t, x \cdot y\right)}{a \cdot 2} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(9\right)\right)}, t, x \cdot y\right)}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(9\right)\right)}, t, x \cdot y\right)}{a \cdot 2} \]

      12. metadata-eval 95.6

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot \color{blue}{-9}, t, x \cdot y\right)}{a \cdot 2} \]

   4. Applied rewrites 95.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 94.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, t \cdot -4.5, y \cdot \frac{x}{a \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}{a \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- (* x y) (* (* z 9.0) t)) (- INFINITY))
   (fma (/ z a) (* t -4.5) (* y (/ x (* a 2.0))))
   (/ (fma (* z -9.0) t (* x y)) (* a 2.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((x * y) - ((z * 9.0) * t)) <= -((double) INFINITY)) {
		tmp = fma((z / a), (t * -4.5), (y * (x / (a * 2.0))));
	} else {
		tmp = fma((z * -9.0), t, (x * y)) / (a * 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) <= Float64(-Inf))
		tmp = fma(Float64(z / a), Float64(t * -4.5), Float64(y * Float64(x / Float64(a * 2.0))));
	else
		tmp = Float64(fma(Float64(z * -9.0), t, Float64(x * y)) / Float64(a * 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(z / a), $MachinePrecision] * N[(t * -4.5), $MachinePrecision] + N[(y * N[(x / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * -9.0), $MachinePrecision] * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t)) < -inf.0

   1. Initial program 66.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{\color{blue}{a \cdot 2}} \]

      5. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} - \frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{a \cdot 2} + \left(\mathsf{neg}\left(\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{a \cdot 2} + \left(\mathsf{neg}\left(\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{a \cdot 2} + \left(\mathsf{neg}\left(\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{a \cdot 2}} + \left(\mathsf{neg}\left(\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\frac{x}{a \cdot 2} \cdot y} + \left(\mathsf{neg}\left(\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{a \cdot 2}, y, \mathsf{neg}\left(\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right)\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{a \cdot 2}}, y, \mathsf{neg}\left(\frac{\left(z \cdot 9\right) \cdot t}{a \cdot 2}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{a \cdot 2}, y, \mathsf{neg}\left(\frac{\left(z \cdot 9\right) \cdot t}{\color{blue}{a \cdot 2}}\right)\right) \]

      14. associate-/l/ N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{a \cdot 2}, y, \mathsf{neg}\left(\color{blue}{\frac{\frac{\left(z \cdot 9\right) \cdot t}{2}}{a}}\right)\right) \]

      15. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{a \cdot 2}, y, \color{blue}{\frac{\frac{\left(z \cdot 9\right) \cdot t}{2}}{\mathsf{neg}\left(a\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{a \cdot 2}, y, \color{blue}{\frac{\frac{\left(z \cdot 9\right) \cdot t}{2}}{\mathsf{neg}\left(a\right)}}\right) \]

   4. Applied rewrites 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{a \cdot 2}, y, \frac{\left(z \cdot t\right) \cdot 4.5}{-a}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{a \cdot 2}} \cdot y + \frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\mathsf{neg}\left(a\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{a \cdot 2}} \cdot y + \frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\mathsf{neg}\left(a\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot 2} \cdot y + \frac{\color{blue}{\left(z \cdot t\right)} \cdot \frac{9}{2}}{\mathsf{neg}\left(a\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{x}{a \cdot 2} \cdot y + \frac{\color{blue}{\left(z \cdot t\right) \cdot \frac{9}{2}}}{\mathsf{neg}\left(a\right)} \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{x}{a \cdot 2} \cdot y + \frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{x}{a \cdot 2} \cdot y + \color{blue}{\frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\mathsf{neg}\left(a\right)}} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\mathsf{neg}\left(a\right)} + \frac{x}{a \cdot 2} \cdot y} \]

      8. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\mathsf{neg}\left(a\right)}} + \frac{x}{a \cdot 2} \cdot y \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot t\right) \cdot \frac{9}{2}}}{\mathsf{neg}\left(a\right)} + \frac{x}{a \cdot 2} \cdot y \]

      10. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\frac{9}{2} \cdot \left(z \cdot t\right)}}{\mathsf{neg}\left(a\right)} + \frac{x}{a \cdot 2} \cdot y \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{\frac{9}{2} \cdot \left(z \cdot t\right)}{\color{blue}{\mathsf{neg}\left(a\right)}} + \frac{x}{a \cdot 2} \cdot y \]

      12. neg-mul-1 N/A

          \[\leadsto \frac{\frac{9}{2} \cdot \left(z \cdot t\right)}{\color{blue}{-1 \cdot a}} + \frac{x}{a \cdot 2} \cdot y \]

      13. times-frac N/A

          \[\leadsto \color{blue}{\frac{\frac{9}{2}}{-1} \cdot \frac{z \cdot t}{a}} + \frac{x}{a \cdot 2} \cdot y \]

      14. metadata-eval N/A

          \[\leadsto \color{blue}{\frac{-9}{2}} \cdot \frac{z \cdot t}{a} + \frac{x}{a \cdot 2} \cdot y \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{-9}{2} \cdot \frac{\color{blue}{z \cdot t}}{a} + \frac{x}{a \cdot 2} \cdot y \]

      16. *-commutative N/A

          \[\leadsto \frac{-9}{2} \cdot \frac{\color{blue}{t \cdot z}}{a} + \frac{x}{a \cdot 2} \cdot y \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} + \frac{x}{a \cdot 2} \cdot y \]

      18. lift-/.f64 N/A

          \[\leadsto \frac{-9}{2} \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) + \frac{x}{a \cdot 2} \cdot y \]

      19. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot t\right) \cdot \frac{z}{a}} + \frac{x}{a \cdot 2} \cdot y \]

      20. *-commutative N/A

          \[\leadsto \color{blue}{\frac{z}{a} \cdot \left(\frac{-9}{2} \cdot t\right)} + \frac{x}{a \cdot 2} \cdot y \]

   6. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{a}, t \cdot -4.5, y \cdot \frac{x}{a \cdot 2}\right)} \]

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z #s(literal 9 binary64)) t))

   1. Initial program 95.6%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. sub-neg N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right) + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 9\right) \cdot t}\right)\right) + x \cdot y}{a \cdot 2} \]

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

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t} + x \cdot y}{a \cdot 2} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z \cdot 9\right), t, x \cdot y\right)}}{a \cdot 2} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right), t, x \cdot y\right)}{a \cdot 2} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(9\right)\right)}, t, x \cdot y\right)}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(9\right)\right)}, t, x \cdot y\right)}{a \cdot 2} \]

      12. metadata-eval 95.6

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot \color{blue}{-9}, t, x \cdot y\right)}{a \cdot 2} \]

   4. Applied rewrites 95.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}}{a \cdot 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -0.0004:\\ \;\;\;\;\frac{t \cdot \left(z \cdot -4.5\right)}{a}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-111}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot 0.5\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4.5}{a} \cdot \left(z \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 -0.0004)
     (/ (* t (* z -4.5)) a)
     (if (<= t_1 5e-111) (/ (* x (* y 0.5)) a) (* (/ -4.5 a) (* z t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -0.0004) {
		tmp = (t * (z * -4.5)) / a;
	} else if (t_1 <= 5e-111) {
		tmp = (x * (y * 0.5)) / a;
	} else {
		tmp = (-4.5 / a) * (z * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if (t_1 <= (-0.0004d0)) then
        tmp = (t * (z * (-4.5d0))) / a
    else if (t_1 <= 5d-111) then
        tmp = (x * (y * 0.5d0)) / a
    else
        tmp = ((-4.5d0) / a) * (z * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -0.0004) {
		tmp = (t * (z * -4.5)) / a;
	} else if (t_1 <= 5e-111) {
		tmp = (x * (y * 0.5)) / a;
	} else {
		tmp = (-4.5 / a) * (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -0.0004:
		tmp = (t * (z * -4.5)) / a
	elif t_1 <= 5e-111:
		tmp = (x * (y * 0.5)) / a
	else:
		tmp = (-4.5 / a) * (z * t)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -0.0004)
		tmp = Float64(Float64(t * Float64(z * -4.5)) / a);
	elseif (t_1 <= 5e-111)
		tmp = Float64(Float64(x * Float64(y * 0.5)) / a);
	else
		tmp = Float64(Float64(-4.5 / a) * Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -0.0004)
		tmp = (t * (z * -4.5)) / a;
	elseif (t_1 <= 5e-111)
		tmp = (x * (y * 0.5)) / a;
	else
		tmp = (-4.5 / a) * (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -0.0004], N[(N[(t * N[(z * -4.5), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[t$95$1, 5e-111], N[(N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-4.5 / a), $MachinePrecision] * N[(z * t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -4.00000000000000019e-4

   1. Initial program 91.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      4. lower-/.f64 73.5

         \[\leadsto -4.5 \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) \]

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{\frac{9}{2}}{-1}} \cdot \left(t \cdot \frac{z}{a}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \color{blue}{\frac{t \cdot z}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \frac{\color{blue}{z \cdot t}}{a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \frac{\color{blue}{z \cdot t}}{a} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{9}{2} \cdot \left(z \cdot t\right)}{-1 \cdot a}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot t\right) \cdot \frac{9}{2}}}{-1 \cdot a} \]

      7. neg-mul-1 N/A

         \[\leadsto \frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{\left(z \cdot t\right) \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(z \cdot t\right)} \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)} \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{z \cdot \left(t \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{z \cdot \left(t \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto z \cdot \color{blue}{\left(t \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)}\right)} \]

      14. metadata-eval N/A

          \[\leadsto z \cdot \left(t \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{-9}{2}\right)}}{\mathsf{neg}\left(a\right)}\right) \]

      15. lift-neg.f64 N/A

          \[\leadsto z \cdot \left(t \cdot \frac{\mathsf{neg}\left(\frac{-9}{2}\right)}{\color{blue}{\mathsf{neg}\left(a\right)}}\right) \]

      16. frac-2neg N/A

          \[\leadsto z \cdot \left(t \cdot \color{blue}{\frac{\frac{-9}{2}}{a}}\right) \]

      17. lower-/.f64 66.8

          \[\leadsto z \cdot \left(t \cdot \color{blue}{\frac{-4.5}{a}}\right) \]

   7. Applied rewrites 66.8%

      \[\leadsto \color{blue}{z \cdot \left(t \cdot \frac{-4.5}{a}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto z \cdot \left(t \cdot \color{blue}{\frac{\frac{-9}{2}}{a}}\right) \]

      2. *-commutative N/A

         \[\leadsto z \cdot \color{blue}{\left(\frac{\frac{-9}{2}}{a} \cdot t\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(z \cdot \frac{\frac{-9}{2}}{a}\right) \cdot t} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{-9}{2}}{a} \cdot z\right)} \cdot t \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{-9}{2}}{a} \cdot z\right) \cdot t} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \frac{\frac{-9}{2}}{a}\right)} \cdot t \]

      7. lift-/.f64 N/A

         \[\leadsto \left(z \cdot \color{blue}{\frac{\frac{-9}{2}}{a}}\right) \cdot t \]

      8. clear-num N/A

         \[\leadsto \left(z \cdot \color{blue}{\frac{1}{\frac{a}{\frac{-9}{2}}}}\right) \cdot t \]

      9. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{z}{\frac{a}{\frac{-9}{2}}}} \cdot t \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{z}{\frac{a}{\frac{-9}{2}}}} \cdot t \]

      11. div-inv N/A

          \[\leadsto \frac{z}{\color{blue}{a \cdot \frac{1}{\frac{-9}{2}}}} \cdot t \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{z}{\color{blue}{a \cdot \frac{1}{\frac{-9}{2}}}} \cdot t \]

      13. metadata-eval 73.3

          \[\leadsto \frac{z}{a \cdot \color{blue}{-0.2222222222222222}} \cdot t \]

   9. Applied rewrites 73.3%

      \[\leadsto \color{blue}{\frac{z}{a \cdot -0.2222222222222222} \cdot t} \]
   10. Step-by-step derivation

       1. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{z}{\frac{-2}{9}}}{a}} \cdot t \]

       2. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{\frac{z}{\frac{-2}{9}} \cdot t}{a}} \]

       3. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{z}{\frac{-2}{9}} \cdot t}{a}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{z}{\frac{-2}{9}} \cdot t}}{a} \]

       5. div-inv N/A

          \[\leadsto \frac{\color{blue}{\left(z \cdot \frac{1}{\frac{-2}{9}}\right)} \cdot t}{a} \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(z \cdot \frac{1}{\frac{-2}{9}}\right)} \cdot t}{a} \]

       7. metadata-eval 77.0

          \[\leadsto \frac{\left(z \cdot \color{blue}{-4.5}\right) \cdot t}{a} \]

   11. Applied rewrites 77.0%

       \[\leadsto \color{blue}{\frac{\left(z \cdot -4.5\right) \cdot t}{a}} \]

   if -4.00000000000000019e-4 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.0000000000000003e-111

   1. Initial program 95.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \frac{1}{2}}}{a} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \frac{1}{2}\right)}}{a} \]

      5. *-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{2} \cdot y\right)}}{a} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{2} \cdot y\right)}}{a} \]

      7. *-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(y \cdot \frac{1}{2}\right)}}{a} \]

      8. lower-*.f64 86.3

         \[\leadsto \frac{x \cdot \color{blue}{\left(y \cdot 0.5\right)}}{a} \]

   5. Applied rewrites 86.3%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot 0.5\right)}{a}} \]

   if 5.0000000000000003e-111 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 90.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      4. lower-/.f64 74.0

         \[\leadsto -4.5 \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) \]

   5. Applied rewrites 74.0%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{\frac{9}{2}}{-1}} \cdot \left(t \cdot \frac{z}{a}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \color{blue}{\frac{t \cdot z}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \frac{\color{blue}{z \cdot t}}{a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \frac{\color{blue}{z \cdot t}}{a} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{9}{2} \cdot \left(z \cdot t\right)}{-1 \cdot a}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot t\right) \cdot \frac{9}{2}}}{-1 \cdot a} \]

      7. neg-mul-1 N/A

         \[\leadsto \frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{\left(z \cdot t\right) \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)} \cdot \left(z \cdot t\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)} \cdot \left(z \cdot t\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{-9}{2}\right)}}{\mathsf{neg}\left(a\right)} \cdot \left(z \cdot t\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{-9}{2}\right)}{\color{blue}{\mathsf{neg}\left(a\right)}} \cdot \left(z \cdot t\right) \]

      14. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{\frac{-9}{2}}{a}} \cdot \left(z \cdot t\right) \]

      15. lower-/.f64 76.3

          \[\leadsto \color{blue}{\frac{-4.5}{a}} \cdot \left(z \cdot t\right) \]

   7. Applied rewrites 76.3%

      \[\leadsto \color{blue}{\frac{-4.5}{a} \cdot \left(z \cdot t\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -0.0004:\\ \;\;\;\;\frac{t \cdot \left(z \cdot -4.5\right)}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{-111}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot 0.5\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4.5}{a} \cdot \left(z \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ t_2 := \frac{-4.5}{a} \cdot \left(z \cdot t\right)\\ \mathbf{if}\;t\_1 \leq -0.0004:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-111}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot 0.5\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)) (t_2 (* (/ -4.5 a) (* z t))))
   (if (<= t_1 -0.0004) t_2 (if (<= t_1 5e-111) (/ (* x (* y 0.5)) a) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double t_2 = (-4.5 / a) * (z * t);
	double tmp;
	if (t_1 <= -0.0004) {
		tmp = t_2;
	} else if (t_1 <= 5e-111) {
		tmp = (x * (y * 0.5)) / a;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    t_2 = ((-4.5d0) / a) * (z * t)
    if (t_1 <= (-0.0004d0)) then
        tmp = t_2
    else if (t_1 <= 5d-111) then
        tmp = (x * (y * 0.5d0)) / a
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double t_2 = (-4.5 / a) * (z * t);
	double tmp;
	if (t_1 <= -0.0004) {
		tmp = t_2;
	} else if (t_1 <= 5e-111) {
		tmp = (x * (y * 0.5)) / a;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	t_2 = (-4.5 / a) * (z * t)
	tmp = 0
	if t_1 <= -0.0004:
		tmp = t_2
	elif t_1 <= 5e-111:
		tmp = (x * (y * 0.5)) / a
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	t_2 = Float64(Float64(-4.5 / a) * Float64(z * t))
	tmp = 0.0
	if (t_1 <= -0.0004)
		tmp = t_2;
	elseif (t_1 <= 5e-111)
		tmp = Float64(Float64(x * Float64(y * 0.5)) / a);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	t_2 = (-4.5 / a) * (z * t);
	tmp = 0.0;
	if (t_1 <= -0.0004)
		tmp = t_2;
	elseif (t_1 <= 5e-111)
		tmp = (x * (y * 0.5)) / a;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.5 / a), $MachinePrecision] * N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.0004], t$95$2, If[LessEqual[t$95$1, 5e-111], N[(N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -4.00000000000000019e-4 or 5.0000000000000003e-111 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 90.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      4. lower-/.f64 73.8

         \[\leadsto -4.5 \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) \]

   5. Applied rewrites 73.8%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{\frac{9}{2}}{-1}} \cdot \left(t \cdot \frac{z}{a}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \color{blue}{\frac{t \cdot z}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \frac{\color{blue}{z \cdot t}}{a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \frac{\color{blue}{z \cdot t}}{a} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{9}{2} \cdot \left(z \cdot t\right)}{-1 \cdot a}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot t\right) \cdot \frac{9}{2}}}{-1 \cdot a} \]

      7. neg-mul-1 N/A

         \[\leadsto \frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{\left(z \cdot t\right) \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)} \cdot \left(z \cdot t\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)} \cdot \left(z \cdot t\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{-9}{2}\right)}}{\mathsf{neg}\left(a\right)} \cdot \left(z \cdot t\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{-9}{2}\right)}{\color{blue}{\mathsf{neg}\left(a\right)}} \cdot \left(z \cdot t\right) \]

      14. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{\frac{-9}{2}}{a}} \cdot \left(z \cdot t\right) \]

      15. lower-/.f64 76.5

          \[\leadsto \color{blue}{\frac{-4.5}{a}} \cdot \left(z \cdot t\right) \]

   7. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{-4.5}{a} \cdot \left(z \cdot t\right)} \]

   if -4.00000000000000019e-4 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.0000000000000003e-111

   1. Initial program 95.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{a}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(x \cdot y\right)}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \frac{1}{2}}}{a} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(y \cdot \frac{1}{2}\right)}}{a} \]

      5. *-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{2} \cdot y\right)}}{a} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{2} \cdot y\right)}}{a} \]

      7. *-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(y \cdot \frac{1}{2}\right)}}{a} \]

      8. lower-*.f64 86.3

         \[\leadsto \frac{x \cdot \color{blue}{\left(y \cdot 0.5\right)}}{a} \]

   5. Applied rewrites 86.3%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot 0.5\right)}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ t_2 := \frac{-4.5}{a} \cdot \left(z \cdot t\right)\\ \mathbf{if}\;t\_1 \leq -0.0004:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-111}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)) (t_2 (* (/ -4.5 a) (* z t))))
   (if (<= t_1 -0.0004) t_2 (if (<= t_1 5e-111) (* (* x y) (/ 0.5 a)) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double t_2 = (-4.5 / a) * (z * t);
	double tmp;
	if (t_1 <= -0.0004) {
		tmp = t_2;
	} else if (t_1 <= 5e-111) {
		tmp = (x * y) * (0.5 / a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    t_2 = ((-4.5d0) / a) * (z * t)
    if (t_1 <= (-0.0004d0)) then
        tmp = t_2
    else if (t_1 <= 5d-111) then
        tmp = (x * y) * (0.5d0 / a)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double t_2 = (-4.5 / a) * (z * t);
	double tmp;
	if (t_1 <= -0.0004) {
		tmp = t_2;
	} else if (t_1 <= 5e-111) {
		tmp = (x * y) * (0.5 / a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	t_2 = (-4.5 / a) * (z * t)
	tmp = 0
	if t_1 <= -0.0004:
		tmp = t_2
	elif t_1 <= 5e-111:
		tmp = (x * y) * (0.5 / a)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	t_2 = Float64(Float64(-4.5 / a) * Float64(z * t))
	tmp = 0.0
	if (t_1 <= -0.0004)
		tmp = t_2;
	elseif (t_1 <= 5e-111)
		tmp = Float64(Float64(x * y) * Float64(0.5 / a));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	t_2 = (-4.5 / a) * (z * t);
	tmp = 0.0;
	if (t_1 <= -0.0004)
		tmp = t_2;
	elseif (t_1 <= 5e-111)
		tmp = (x * y) * (0.5 / a);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-4.5 / a), $MachinePrecision] * N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.0004], t$95$2, If[LessEqual[t$95$1, 5e-111], N[(N[(x * y), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -4.00000000000000019e-4 or 5.0000000000000003e-111 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 90.8%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      4. lower-/.f64 73.8

         \[\leadsto -4.5 \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) \]

   5. Applied rewrites 73.8%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{\frac{9}{2}}{-1}} \cdot \left(t \cdot \frac{z}{a}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \color{blue}{\frac{t \cdot z}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \frac{\color{blue}{z \cdot t}}{a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \frac{\color{blue}{z \cdot t}}{a} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{9}{2} \cdot \left(z \cdot t\right)}{-1 \cdot a}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot t\right) \cdot \frac{9}{2}}}{-1 \cdot a} \]

      7. neg-mul-1 N/A

         \[\leadsto \frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{\left(z \cdot t\right) \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)}} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)} \cdot \left(z \cdot t\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)} \cdot \left(z \cdot t\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\frac{-9}{2}\right)}}{\mathsf{neg}\left(a\right)} \cdot \left(z \cdot t\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{-9}{2}\right)}{\color{blue}{\mathsf{neg}\left(a\right)}} \cdot \left(z \cdot t\right) \]

      14. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{\frac{-9}{2}}{a}} \cdot \left(z \cdot t\right) \]

      15. lower-/.f64 76.5

          \[\leadsto \color{blue}{\frac{-4.5}{a}} \cdot \left(z \cdot t\right) \]

   7. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{-4.5}{a} \cdot \left(z \cdot t\right)} \]

   if -4.00000000000000019e-4 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.0000000000000003e-111

   1. Initial program 95.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{\color{blue}{a \cdot 2}} \]

      6. div-inv N/A

         \[\leadsto \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}} \]

      8. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \cdot \frac{1}{a \cdot 2} \]

      9. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right)} \cdot \frac{1}{a \cdot 2} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right) + x \cdot y\right)} \cdot \frac{1}{a \cdot 2} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 9\right) \cdot t}\right)\right) + x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 9\right)} \cdot t\right)\right) + x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      13. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot \left(9 \cdot t\right)}\right)\right) + x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

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

          \[\leadsto \left(\color{blue}{z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)} + x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{neg}\left(9 \cdot t\right), x \cdot y\right)} \cdot \frac{1}{a \cdot 2} \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{neg}\left(\color{blue}{t \cdot 9}\right), x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

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

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{t \cdot \left(\mathsf{neg}\left(9\right)\right)}, x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{t \cdot \left(\mathsf{neg}\left(9\right)\right)}, x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, t \cdot \color{blue}{-9}, x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      20. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{\color{blue}{a \cdot 2}} \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{\color{blue}{2 \cdot a}} \]

      22. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}} \]

      23. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}} \]

      24. metadata-eval 95.4

          \[\leadsto \mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   4. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{0.5}{a}} \]

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 86.1

         \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{0.5}{a} \]

   7. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{0.5}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t\_1 \leq -0.0004:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-111}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (* z 9.0) t)))
   (if (<= t_1 -0.0004)
     (* -4.5 (* t (/ z a)))
     (if (<= t_1 5e-111) (* (* x y) (/ 0.5 a)) (* t (* (/ z a) -4.5))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -0.0004) {
		tmp = -4.5 * (t * (z / a));
	} else if (t_1 <= 5e-111) {
		tmp = (x * y) * (0.5 / a);
	} else {
		tmp = t * ((z / a) * -4.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * 9.0d0) * t
    if (t_1 <= (-0.0004d0)) then
        tmp = (-4.5d0) * (t * (z / a))
    else if (t_1 <= 5d-111) then
        tmp = (x * y) * (0.5d0 / a)
    else
        tmp = t * ((z / a) * (-4.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * 9.0) * t;
	double tmp;
	if (t_1 <= -0.0004) {
		tmp = -4.5 * (t * (z / a));
	} else if (t_1 <= 5e-111) {
		tmp = (x * y) * (0.5 / a);
	} else {
		tmp = t * ((z / a) * -4.5);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z * 9.0) * t
	tmp = 0
	if t_1 <= -0.0004:
		tmp = -4.5 * (t * (z / a))
	elif t_1 <= 5e-111:
		tmp = (x * y) * (0.5 / a)
	else:
		tmp = t * ((z / a) * -4.5)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * 9.0) * t)
	tmp = 0.0
	if (t_1 <= -0.0004)
		tmp = Float64(-4.5 * Float64(t * Float64(z / a)));
	elseif (t_1 <= 5e-111)
		tmp = Float64(Float64(x * y) * Float64(0.5 / a));
	else
		tmp = Float64(t * Float64(Float64(z / a) * -4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * 9.0) * t;
	tmp = 0.0;
	if (t_1 <= -0.0004)
		tmp = -4.5 * (t * (z / a));
	elseif (t_1 <= 5e-111)
		tmp = (x * y) * (0.5 / a);
	else
		tmp = t * ((z / a) * -4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t$95$1, -0.0004], N[(-4.5 * N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-111], N[(N[(x * y), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < -4.00000000000000019e-4

   1. Initial program 91.1%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      4. lower-/.f64 73.5

         \[\leadsto -4.5 \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) \]

   5. Applied rewrites 73.5%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]

   if -4.00000000000000019e-4 < (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.0000000000000003e-111

   1. Initial program 95.7%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{\color{blue}{a \cdot 2}} \]

      6. div-inv N/A

         \[\leadsto \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}} \]

      8. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \cdot \frac{1}{a \cdot 2} \]

      9. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right)} \cdot \frac{1}{a \cdot 2} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right) + x \cdot y\right)} \cdot \frac{1}{a \cdot 2} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 9\right) \cdot t}\right)\right) + x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 9\right)} \cdot t\right)\right) + x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      13. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot \left(9 \cdot t\right)}\right)\right) + x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

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

          \[\leadsto \left(\color{blue}{z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)} + x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{neg}\left(9 \cdot t\right), x \cdot y\right)} \cdot \frac{1}{a \cdot 2} \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{neg}\left(\color{blue}{t \cdot 9}\right), x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

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

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{t \cdot \left(\mathsf{neg}\left(9\right)\right)}, x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{t \cdot \left(\mathsf{neg}\left(9\right)\right)}, x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, t \cdot \color{blue}{-9}, x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      20. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{\color{blue}{a \cdot 2}} \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{\color{blue}{2 \cdot a}} \]

      22. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}} \]

      23. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}} \]

      24. metadata-eval 95.4

          \[\leadsto \mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   4. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{0.5}{a}} \]

   5. Taylor expanded in z around 0

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

      1. lower-*.f64 86.1

         \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{0.5}{a} \]

   7. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\left(x \cdot y\right)} \cdot \frac{0.5}{a} \]

   if 5.0000000000000003e-111 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 90.5%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      4. lower-/.f64 74.0

         \[\leadsto -4.5 \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) \]

   5. Applied rewrites 74.0%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-9}{2} \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(\frac{z}{a} \cdot t\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t} \]

      5. lower-*.f64 74.2

         \[\leadsto \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right)} \cdot t \]

   7. Applied rewrites 74.2%

      \[\leadsto \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right) \cdot t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq -0.0004:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{-111}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* (* z 9.0) t) 5e+291)
   (/ (fma (* z -9.0) t (* x y)) (* a 2.0))
   (* t (* (/ z a) -4.5))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * 9.0) * t) <= 5e+291) {
		tmp = fma((z * -9.0), t, (x * y)) / (a * 2.0);
	} else {
		tmp = t * ((z / a) * -4.5);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z * 9.0) * t) <= 5e+291)
		tmp = Float64(fma(Float64(z * -9.0), t, Float64(x * y)) / Float64(a * 2.0));
	else
		tmp = Float64(t * Float64(Float64(z / a) * -4.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision], 5e+291], N[(N[(N[(z * -9.0), $MachinePrecision] * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.0000000000000001e291

   1. Initial program 95.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. sub-neg N/A

         \[\leadsto \frac{\color{blue}{x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)}}{a \cdot 2} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right) + x \cdot y}}{a \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 9\right) \cdot t}\right)\right) + x \cdot y}{a \cdot 2} \]

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

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(z \cdot 9\right)\right) \cdot t} + x \cdot y}{a \cdot 2} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(z \cdot 9\right), t, x \cdot y\right)}}{a \cdot 2} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{z \cdot 9}\right), t, x \cdot y\right)}{a \cdot 2} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(9\right)\right)}, t, x \cdot y\right)}{a \cdot 2} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{z \cdot \left(\mathsf{neg}\left(9\right)\right)}, t, x \cdot y\right)}{a \cdot 2} \]

      12. metadata-eval 95.0

          \[\leadsto \frac{\mathsf{fma}\left(z \cdot \color{blue}{-9}, t, x \cdot y\right)}{a \cdot 2} \]

   4. Applied rewrites 95.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}}{a \cdot 2} \]

   if 5.0000000000000001e291 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 58.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      4. lower-/.f64 93.7

         \[\leadsto -4.5 \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) \]

   5. Applied rewrites 93.7%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-9}{2} \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(\frac{z}{a} \cdot t\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t} \]

      5. lower-*.f64 93.8

         \[\leadsto \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right)} \cdot t \]

   7. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right) \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z \cdot -9, t, x \cdot y\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (* (* z 9.0) t) 5e+291)
   (* (fma z (* t -9.0) (* x y)) (/ 0.5 a))
   (* t (* (/ z a) -4.5))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((z * 9.0) * t) <= 5e+291) {
		tmp = fma(z, (t * -9.0), (x * y)) * (0.5 / a);
	} else {
		tmp = t * ((z / a) * -4.5);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(z * 9.0) * t) <= 5e+291)
		tmp = Float64(fma(z, Float64(t * -9.0), Float64(x * y)) * Float64(0.5 / a));
	else
		tmp = Float64(t * Float64(Float64(z / a) * -4.5));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision], 5e+291], N[(N[(z * N[(t * -9.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(t * N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z #s(literal 9 binary64)) t) < 5.0000000000000001e291

   1. Initial program 95.0%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y} - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right)} \cdot t}{a \cdot 2} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \color{blue}{\left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y - \left(z \cdot 9\right) \cdot t}}{a \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{\color{blue}{a \cdot 2}} \]

      6. div-inv N/A

         \[\leadsto \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right) \cdot \frac{1}{a \cdot 2}} \]

      8. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y - \left(z \cdot 9\right) \cdot t\right)} \cdot \frac{1}{a \cdot 2} \]

      9. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right)\right)} \cdot \frac{1}{a \cdot 2} \]

      10. +-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\left(z \cdot 9\right) \cdot t\right)\right) + x \cdot y\right)} \cdot \frac{1}{a \cdot 2} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 9\right) \cdot t}\right)\right) + x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(z \cdot 9\right)} \cdot t\right)\right) + x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      13. associate-*l* N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot \left(9 \cdot t\right)}\right)\right) + x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

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

          \[\leadsto \left(\color{blue}{z \cdot \left(\mathsf{neg}\left(9 \cdot t\right)\right)} + x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      15. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{neg}\left(9 \cdot t\right), x \cdot y\right)} \cdot \frac{1}{a \cdot 2} \]

      16. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, \mathsf{neg}\left(\color{blue}{t \cdot 9}\right), x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

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

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{t \cdot \left(\mathsf{neg}\left(9\right)\right)}, x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, \color{blue}{t \cdot \left(\mathsf{neg}\left(9\right)\right)}, x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      19. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, t \cdot \color{blue}{-9}, x \cdot y\right) \cdot \frac{1}{a \cdot 2} \]

      20. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{\color{blue}{a \cdot 2}} \]

      21. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{1}{\color{blue}{2 \cdot a}} \]

      22. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}} \]

      23. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}} \]

      24. metadata-eval 94.5

          \[\leadsto \mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   4. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{0.5}{a}} \]

   if 5.0000000000000001e291 < (*.f64 (*.f64 z #s(literal 9 binary64)) t)

   1. Initial program 58.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      4. lower-/.f64 93.7

         \[\leadsto -4.5 \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) \]

   5. Applied rewrites 93.7%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-9}{2} \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(\frac{z}{a} \cdot t\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t} \]

      5. lower-*.f64 93.8

         \[\leadsto \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right)} \cdot t \]

   7. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right) \cdot t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \leq 5 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(z, t \cdot -9, x \cdot y\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-308}:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t \cdot \frac{-4.5}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x 5e-308) (* t (* (/ z a) -4.5)) (* z (* t (/ -4.5 a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= 5e-308) {
		tmp = t * ((z / a) * -4.5);
	} else {
		tmp = z * (t * (-4.5 / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= 5d-308) then
        tmp = t * ((z / a) * (-4.5d0))
    else
        tmp = z * (t * ((-4.5d0) / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= 5e-308) {
		tmp = t * ((z / a) * -4.5);
	} else {
		tmp = z * (t * (-4.5 / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= 5e-308:
		tmp = t * ((z / a) * -4.5)
	else:
		tmp = z * (t * (-4.5 / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= 5e-308)
		tmp = Float64(t * Float64(Float64(z / a) * -4.5));
	else
		tmp = Float64(z * Float64(t * Float64(-4.5 / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= 5e-308)
		tmp = t * ((z / a) * -4.5);
	else
		tmp = z * (t * (-4.5 / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, 5e-308], N[(t * N[(N[(z / a), $MachinePrecision] * -4.5), $MachinePrecision]), $MachinePrecision], N[(z * N[(t * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999955e-308

   1. Initial program 95.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      4. lower-/.f64 50.7

         \[\leadsto -4.5 \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) \]

   5. Applied rewrites 50.7%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{-9}{2} \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(\frac{z}{a} \cdot t\right)} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-9}{2} \cdot \frac{z}{a}\right) \cdot t} \]

      5. lower-*.f64 50.8

         \[\leadsto \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right)} \cdot t \]

   7. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\left(-4.5 \cdot \frac{z}{a}\right) \cdot t} \]

   if 4.99999999999999955e-308 < x

   1. Initial program 90.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      4. lower-/.f64 50.4

         \[\leadsto -4.5 \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) \]

   5. Applied rewrites 50.4%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{\frac{9}{2}}{-1}} \cdot \left(t \cdot \frac{z}{a}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \color{blue}{\frac{t \cdot z}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \frac{\color{blue}{z \cdot t}}{a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \frac{\color{blue}{z \cdot t}}{a} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{9}{2} \cdot \left(z \cdot t\right)}{-1 \cdot a}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot t\right) \cdot \frac{9}{2}}}{-1 \cdot a} \]

      7. neg-mul-1 N/A

         \[\leadsto \frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{\left(z \cdot t\right) \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(z \cdot t\right)} \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)} \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{z \cdot \left(t \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{z \cdot \left(t \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto z \cdot \color{blue}{\left(t \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)}\right)} \]

      14. metadata-eval N/A

          \[\leadsto z \cdot \left(t \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{-9}{2}\right)}}{\mathsf{neg}\left(a\right)}\right) \]

      15. lift-neg.f64 N/A

          \[\leadsto z \cdot \left(t \cdot \frac{\mathsf{neg}\left(\frac{-9}{2}\right)}{\color{blue}{\mathsf{neg}\left(a\right)}}\right) \]

      16. frac-2neg N/A

          \[\leadsto z \cdot \left(t \cdot \color{blue}{\frac{\frac{-9}{2}}{a}}\right) \]

      17. lower-/.f64 47.7

          \[\leadsto z \cdot \left(t \cdot \color{blue}{\frac{-4.5}{a}}\right) \]

   7. Applied rewrites 47.7%

      \[\leadsto \color{blue}{z \cdot \left(t \cdot \frac{-4.5}{a}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-308}:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t \cdot \frac{-4.5}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-307}:\\ \;\;\;\;-4.5 \cdot \left(t \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t \cdot \frac{-4.5}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -1.75e-307) (* -4.5 (* t (/ z a))) (* z (* t (/ -4.5 a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.75e-307) {
		tmp = -4.5 * (t * (z / a));
	} else {
		tmp = z * (t * (-4.5 / a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= (-1.75d-307)) then
        tmp = (-4.5d0) * (t * (z / a))
    else
        tmp = z * (t * ((-4.5d0) / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.75e-307) {
		tmp = -4.5 * (t * (z / a));
	} else {
		tmp = z * (t * (-4.5 / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -1.75e-307:
		tmp = -4.5 * (t * (z / a))
	else:
		tmp = z * (t * (-4.5 / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -1.75e-307)
		tmp = Float64(-4.5 * Float64(t * Float64(z / a)));
	else
		tmp = Float64(z * Float64(t * Float64(-4.5 / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -1.75e-307)
		tmp = -4.5 * (t * (z / a));
	else
		tmp = z * (t * (-4.5 / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -1.75e-307], N[(-4.5 * N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(t * N[(-4.5 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7500000000000001e-307

   1. Initial program 95.9%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      4. lower-/.f64 50.7

         \[\leadsto -4.5 \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) \]

   5. Applied rewrites 50.7%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]

   if -1.7500000000000001e-307 < x

   1. Initial program 90.3%

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

      4. lower-/.f64 50.4

         \[\leadsto -4.5 \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) \]

   5. Applied rewrites 50.4%

      \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{\frac{9}{2}}{-1}} \cdot \left(t \cdot \frac{z}{a}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \color{blue}{\frac{t \cdot z}{a}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \frac{\color{blue}{z \cdot t}}{a} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{9}{2}}{-1} \cdot \frac{\color{blue}{z \cdot t}}{a} \]

      5. times-frac N/A

         \[\leadsto \color{blue}{\frac{\frac{9}{2} \cdot \left(z \cdot t\right)}{-1 \cdot a}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z \cdot t\right) \cdot \frac{9}{2}}}{-1 \cdot a} \]

      7. neg-mul-1 N/A

         \[\leadsto \frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\left(z \cdot t\right) \cdot \frac{9}{2}}{\color{blue}{\mathsf{neg}\left(a\right)}} \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{\left(z \cdot t\right) \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \color{blue}{\left(z \cdot t\right)} \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)} \]

      11. associate-*l* N/A

          \[\leadsto \color{blue}{z \cdot \left(t \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)}\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{z \cdot \left(t \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto z \cdot \color{blue}{\left(t \cdot \frac{\frac{9}{2}}{\mathsf{neg}\left(a\right)}\right)} \]

      14. metadata-eval N/A

          \[\leadsto z \cdot \left(t \cdot \frac{\color{blue}{\mathsf{neg}\left(\frac{-9}{2}\right)}}{\mathsf{neg}\left(a\right)}\right) \]

      15. lift-neg.f64 N/A

          \[\leadsto z \cdot \left(t \cdot \frac{\mathsf{neg}\left(\frac{-9}{2}\right)}{\color{blue}{\mathsf{neg}\left(a\right)}}\right) \]

      16. frac-2neg N/A

          \[\leadsto z \cdot \left(t \cdot \color{blue}{\frac{\frac{-9}{2}}{a}}\right) \]

      17. lower-/.f64 47.7

          \[\leadsto z \cdot \left(t \cdot \color{blue}{\frac{-4.5}{a}}\right) \]

   7. Applied rewrites 47.7%

      \[\leadsto \color{blue}{z \cdot \left(t \cdot \frac{-4.5}{a}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 51.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ -4.5 \cdot \left(t \cdot \frac{z}{a}\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (* -4.5 (* t (/ z a))))

C

double code(double x, double y, double z, double t, double a) {
	return -4.5 * (t * (z / a));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = (-4.5d0) * (t * (z / a))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return -4.5 * (t * (z / a));
}

Python

def code(x, y, z, t, a):
	return -4.5 * (t * (z / a))

Julia

function code(x, y, z, t, a)
	return Float64(-4.5 * Float64(t * Float64(z / a)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = -4.5 * (t * (z / a));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(-4.5 * N[(t * N[(z / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.0%

   \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-9}{2} \cdot \frac{t \cdot z}{a}} \]

   2. associate-/l* N/A

      \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{-9}{2} \cdot \color{blue}{\left(t \cdot \frac{z}{a}\right)} \]

   4. lower-/.f64 50.6

      \[\leadsto -4.5 \cdot \left(t \cdot \color{blue}{\frac{z}{a}}\right) \]

5. Applied rewrites 50.6%

   \[\leadsto \color{blue}{-4.5 \cdot \left(t \cdot \frac{z}{a}\right)} \]
6. Add Preprocessing

Developer Target 1: 93.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< a -2.090464557976709e+86)
   (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z))))
   (if (< a 2.144030707833976e+99)
     (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0))
     (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a < (-2.090464557976709d+86)) then
        tmp = (0.5d0 * ((y * x) / a)) - (4.5d0 * (t / (a / z)))
    else if (a < 2.144030707833976d+99) then
        tmp = ((x * y) - (z * (9.0d0 * t))) / (a * 2.0d0)
    else
        tmp = ((y / a) * (x * 0.5d0)) - ((t / a) * (z * 4.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a < -2.090464557976709e+86) {
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	} else if (a < 2.144030707833976e+99) {
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	} else {
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a < -2.090464557976709e+86:
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)))
	elif a < 2.144030707833976e+99:
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0)
	else:
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a < -2.090464557976709e+86)
		tmp = Float64(Float64(0.5 * Float64(Float64(y * x) / a)) - Float64(4.5 * Float64(t / Float64(a / z))));
	elseif (a < 2.144030707833976e+99)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(9.0 * t))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(Float64(y / a) * Float64(x * 0.5)) - Float64(Float64(t / a) * Float64(z * 4.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a < -2.090464557976709e+86)
		tmp = (0.5 * ((y * x) / a)) - (4.5 * (t / (a / z)));
	elseif (a < 2.144030707833976e+99)
		tmp = ((x * y) - (z * (9.0 * t))) / (a * 2.0);
	else
		tmp = ((y / a) * (x * 0.5)) - ((t / a) * (z * 4.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[a, -2.090464557976709e+86], N[(N[(0.5 * N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] - N[(4.5 * N[(t / N[(a / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[a, 2.144030707833976e+99], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(9.0 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / a), $MachinePrecision] * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[(N[(t / a), $MachinePrecision] * N[(z * 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024219 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< a -209046455797670900000000000000000000000000000000000000000000000000000000000000000000000) (- (* 1/2 (/ (* y x) a)) (* 9/2 (/ t (/ a z)))) (if (< a 2144030707833976000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* x y) (* z (* 9 t))) (* a 2)) (- (* (/ y a) (* x 1/2)) (* (/ t a) (* z 9/2))))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))