Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A

Percentage Accurate: 84.9% → 95.9%

Time: 10.3s

Alternatives: 12

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

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)) / (t - (a * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) / (t - (a * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot z\\ t_2 := \frac{t\_1}{t - z \cdot a}\\ t_3 := \frac{t\_1}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(z, a, -t\right)}\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-299}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{t\_1}{a}}{\frac{t}{a} - z}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y z)))
        (t_2 (/ t_1 (- t (* z a))))
        (t_3 (/ t_1 (fma (- z) a t))))
   (if (<= t_2 (- INFINITY))
     (* z (/ y (fma z a (- t))))
     (if (<= t_2 -4e-299)
       t_3
       (if (<= t_2 0.0)
         (/ (/ t_1 a) (- (/ t a) z))
         (if (<= t_2 2e+303) t_3 (/ (- y (/ x z)) a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = t_1 / (t - (z * a));
	double t_3 = t_1 / fma(-z, a, t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = z * (y / fma(z, a, -t));
	} else if (t_2 <= -4e-299) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = (t_1 / a) / ((t / a) - z);
	} else if (t_2 <= 2e+303) {
		tmp = t_3;
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y * z))
	t_2 = Float64(t_1 / Float64(t - Float64(z * a)))
	t_3 = Float64(t_1 / fma(Float64(-z), a, t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(z * Float64(y / fma(z, a, Float64(-t))));
	elseif (t_2 <= -4e-299)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(t_1 / a) / Float64(Float64(t / a) - z));
	elseif (t_2 <= 2e+303)
		tmp = t_3;
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(z * N[(y / N[(z * a + (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -4e-299], t$95$3, If[LessEqual[t$95$2, 0.0], N[(N[(t$95$1 / a), $MachinePrecision] / N[(N[(t / a), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+303], t$95$3, N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0

   1. Initial program 43.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 28.7

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

   5. Applied rewrites 28.7%

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

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

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{0 - \left(t - a \cdot z\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y \cdot z}{0 - \color{blue}{\left(t - a \cdot z\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{y \cdot z}{0 - \left(t - \color{blue}{a \cdot z}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{y \cdot z}{0 - \left(t - \color{blue}{z \cdot a}\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{y \cdot z}{0 - \left(t - \color{blue}{z \cdot a}\right)} \]

      13. associate-+l- N/A

          \[\leadsto \frac{y \cdot z}{\color{blue}{\left(0 - t\right) + z \cdot a}} \]

      14. neg-sub0 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/l* N/A

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

      22. lower-*.f64 N/A

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

      23. lower-/.f64 89.8

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

   7. Applied rewrites 89.8%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -3.99999999999999997e-299 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2e303

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if -3.99999999999999997e-299 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 72.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 72.6

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

   5. Applied rewrites 72.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if 2e303 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 45.4%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 88.2

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

   8. Applied rewrites 88.2%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(z, a, -t\right)}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -4 \cdot 10^{-299}:\\ \;\;\;\;\frac{x - y \cdot z}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{\frac{x - y \cdot z}{a}}{\frac{t}{a} - z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{x - y \cdot z}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot z\\ t_2 := t - z \cdot a\\ t_3 := \frac{t\_1}{t\_2}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{-254}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, -t\right)}, \frac{x}{t\_2}\right)\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\frac{t\_1}{a}}{\frac{t}{a} - z}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y z))) (t_2 (- t (* z a))) (t_3 (/ t_1 t_2)))
   (if (<= t_3 -1e-254)
     (fma y (/ z (fma z a (- t))) (/ x t_2))
     (if (<= t_3 0.0)
       (/ (/ t_1 a) (- (/ t a) z))
       (if (<= t_3 2e+303) (/ t_1 (fma (- z) a t)) (/ (- y (/ x z)) a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = t - (z * a);
	double t_3 = t_1 / t_2;
	double tmp;
	if (t_3 <= -1e-254) {
		tmp = fma(y, (z / fma(z, a, -t)), (x / t_2));
	} else if (t_3 <= 0.0) {
		tmp = (t_1 / a) / ((t / a) - z);
	} else if (t_3 <= 2e+303) {
		tmp = t_1 / fma(-z, a, t);
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y * z))
	t_2 = Float64(t - Float64(z * a))
	t_3 = Float64(t_1 / t_2)
	tmp = 0.0
	if (t_3 <= -1e-254)
		tmp = fma(y, Float64(z / fma(z, a, Float64(-t))), Float64(x / t_2));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(t_1 / a) / Float64(Float64(t / a) - z));
	elseif (t_3 <= 2e+303)
		tmp = Float64(t_1 / fma(Float64(-z), a, t));
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-254], N[(y * N[(z / N[(z * a + (-t)), $MachinePrecision]), $MachinePrecision] + N[(x / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(t$95$1 / a), $MachinePrecision] / N[(N[(t / a), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+303], N[(t$95$1 / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -9.9999999999999991e-255

   1. Initial program 87.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 96.5%

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

   if -9.9999999999999991e-255 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 75.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 73.6

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

   5. Applied rewrites 73.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.7

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

   7. Applied rewrites 97.7%

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

   if 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2e303

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if 2e303 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 45.4%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 88.2

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

   8. Applied rewrites 88.2%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -1 \cdot 10^{-254}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{\mathsf{fma}\left(z, a, -t\right)}, \frac{x}{t - z \cdot a}\right)\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{\frac{x - y \cdot z}{a}}{\frac{t}{a} - z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{x - y \cdot z}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot z\\ t_2 := \frac{t\_1}{t - z \cdot a}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(z, a, -t\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y z))) (t_2 (/ t_1 (- t (* z a)))))
   (if (<= t_2 (- INFINITY))
     (* z (/ y (fma z a (- t))))
     (if (<= t_2 2e+303) (/ t_1 (fma (- z) a t)) (/ (- y (/ x z)) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = t_1 / (t - (z * a));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = z * (y / fma(z, a, -t));
	} else if (t_2 <= 2e+303) {
		tmp = t_1 / fma(-z, a, t);
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y * z))
	t_2 = Float64(t_1 / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(z * Float64(y / fma(z, a, Float64(-t))));
	elseif (t_2 <= 2e+303)
		tmp = Float64(t_1 / fma(Float64(-z), a, t));
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(z * N[(y / N[(z * a + (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+303], N[(t$95$1 / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0

   1. Initial program 43.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 28.7

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

   5. Applied rewrites 28.7%

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

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

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{0 - \left(t - a \cdot z\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y \cdot z}{0 - \color{blue}{\left(t - a \cdot z\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{y \cdot z}{0 - \left(t - \color{blue}{a \cdot z}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{y \cdot z}{0 - \left(t - \color{blue}{z \cdot a}\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{y \cdot z}{0 - \left(t - \color{blue}{z \cdot a}\right)} \]

      13. associate-+l- N/A

          \[\leadsto \frac{y \cdot z}{\color{blue}{\left(0 - t\right) + z \cdot a}} \]

      14. neg-sub0 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/l* N/A

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

      22. lower-*.f64 N/A

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

      23. lower-/.f64 89.8

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

   7. Applied rewrites 89.8%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2e303

   1. Initial program 94.6%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 94.6

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

   4. Applied rewrites 94.6%

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

   if 2e303 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 45.4%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 88.2

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

   8. Applied rewrites 88.2%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(z, a, -t\right)}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{x - y \cdot z}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(z, a, -t\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_1 (- INFINITY))
     (* z (/ y (fma z a (- t))))
     (if (<= t_1 2e+303) t_1 (/ (- y (/ x z)) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = z * (y / fma(z, a, -t));
	} else if (t_1 <= 2e+303) {
		tmp = t_1;
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(z * Float64(y / fma(z, a, Float64(-t))));
	elseif (t_1 <= 2e+303)
		tmp = t_1;
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(z * N[(y / N[(z * a + (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+303], t$95$1, N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0

   1. Initial program 43.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 28.7

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

   5. Applied rewrites 28.7%

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

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

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{0 - \left(t - a \cdot z\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y \cdot z}{0 - \color{blue}{\left(t - a \cdot z\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{y \cdot z}{0 - \left(t - \color{blue}{a \cdot z}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{y \cdot z}{0 - \left(t - \color{blue}{z \cdot a}\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{y \cdot z}{0 - \left(t - \color{blue}{z \cdot a}\right)} \]

      13. associate-+l- N/A

          \[\leadsto \frac{y \cdot z}{\color{blue}{\left(0 - t\right) + z \cdot a}} \]

      14. neg-sub0 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/l* N/A

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

      22. lower-*.f64 N/A

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

      23. lower-/.f64 89.8

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

   7. Applied rewrites 89.8%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2e303

   1. Initial program 94.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   if 2e303 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 45.4%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 88.2

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

   8. Applied rewrites 88.2%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(z, a, -t\right)}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z}{z \cdot a - t}\\ \mathbf{if}\;z \leq -3 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-223}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ z (- (* z a) t)))))
   (if (<= z -3e-30)
     t_1
     (if (<= z -2.1e-223)
       (/ x (- t (* z a)))
       (if (<= z 9.2e-75) (/ (- x (* y z)) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z / ((z * a) - t));
	double tmp;
	if (z <= -3e-30) {
		tmp = t_1;
	} else if (z <= -2.1e-223) {
		tmp = x / (t - (z * a));
	} else if (z <= 9.2e-75) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = t_1;
	}
	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 = y * (z / ((z * a) - t))
    if (z <= (-3d-30)) then
        tmp = t_1
    else if (z <= (-2.1d-223)) then
        tmp = x / (t - (z * a))
    else if (z <= 9.2d-75) then
        tmp = (x - (y * z)) / t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (z / ((z * a) - t));
	double tmp;
	if (z <= -3e-30) {
		tmp = t_1;
	} else if (z <= -2.1e-223) {
		tmp = x / (t - (z * a));
	} else if (z <= 9.2e-75) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * (z / ((z * a) - t))
	tmp = 0
	if z <= -3e-30:
		tmp = t_1
	elif z <= -2.1e-223:
		tmp = x / (t - (z * a))
	elif z <= 9.2e-75:
		tmp = (x - (y * z)) / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(z / Float64(Float64(z * a) - t)))
	tmp = 0.0
	if (z <= -3e-30)
		tmp = t_1;
	elseif (z <= -2.1e-223)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 9.2e-75)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (z / ((z * a) - t));
	tmp = 0.0;
	if (z <= -3e-30)
		tmp = t_1;
	elseif (z <= -2.1e-223)
		tmp = x / (t - (z * a));
	elseif (z <= 9.2e-75)
		tmp = (x - (y * z)) / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3e-30], t$95$1, If[LessEqual[z, -2.1e-223], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9.2e-75], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.9999999999999999e-30 or 9.2e-75 < z

   1. Initial program 75.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 55.3

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

   8. Applied rewrites 55.3%

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

   9. Taylor expanded in y around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower--.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 66.9

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

   11. Applied rewrites 66.9%

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

   if -2.9999999999999999e-30 < z < -2.09999999999999982e-223

   1. Initial program 99.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if -2.09999999999999982e-223 < z < 9.2e-75

   1. Initial program 99.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

Alternative 6: 67.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{+107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= a -3.8e+107) t_1 (if (<= a 8.2e-73) (/ (fma (- z) y x) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (a <= -3.8e+107) {
		tmp = t_1;
	} else if (a <= 8.2e-73) {
		tmp = fma(-z, y, x) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - Float64(x / z)) / a)
	tmp = 0.0
	if (a <= -3.8e+107)
		tmp = t_1;
	elseif (a <= 8.2e-73)
		tmp = Float64(fma(Float64(-z), y, x) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[a, -3.8e+107], t$95$1, If[LessEqual[a, 8.2e-73], N[(N[((-z) * y + x), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.7999999999999998e107 or 8.20000000000000032e-73 < a

   1. Initial program 76.4%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 74.7

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

   8. Applied rewrites 74.7%

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

   if -3.7999999999999998e107 < a < 8.20000000000000032e-73

   1. Initial program 94.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. div-sub N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 73.8

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

   8. Applied rewrites 73.8%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-neg.f64 N/A

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

      7. lower-fma.f64 73.8

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

   10. Applied rewrites 73.8%

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

Alternative 7: 66.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;z \cdot \frac{y}{\mathsf{fma}\left(z, a, -t\right)}\\ \mathbf{elif}\;y \leq 55000000000000:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -5.5e-5)
   (* z (/ y (fma z a (- t))))
   (if (<= y 55000000000000.0) (/ x (- t (* z a))) (* y (/ z (- (* z a) t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -5.5e-5) {
		tmp = z * (y / fma(z, a, -t));
	} else if (y <= 55000000000000.0) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y * (z / ((z * a) - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -5.5e-5)
		tmp = Float64(z * Float64(y / fma(z, a, Float64(-t))));
	elseif (y <= 55000000000000.0)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y * Float64(z / Float64(Float64(z * a) - t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -5.5e-5], N[(z * N[(y / N[(z * a + (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 55000000000000.0], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(z / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.5000000000000002e-5

   1. Initial program 78.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 53.0

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

   5. Applied rewrites 53.0%

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

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

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \frac{y \cdot z}{\color{blue}{0 - \left(t - a \cdot z\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{y \cdot z}{0 - \color{blue}{\left(t - a \cdot z\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{y \cdot z}{0 - \left(t - \color{blue}{a \cdot z}\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{y \cdot z}{0 - \left(t - \color{blue}{z \cdot a}\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{y \cdot z}{0 - \left(t - \color{blue}{z \cdot a}\right)} \]

      13. associate-+l- N/A

          \[\leadsto \frac{y \cdot z}{\color{blue}{\left(0 - t\right) + z \cdot a}} \]

      14. neg-sub0 N/A

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

      15. lift-neg.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/l* N/A

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

      22. lower-*.f64 N/A

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

      23. lower-/.f64 64.4

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

   7. Applied rewrites 64.4%

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

   if -5.5000000000000002e-5 < y < 5.5e13

   1. Initial program 94.4%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if 5.5e13 < y

   1. Initial program 76.4%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 56.3

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

   8. Applied rewrites 56.3%

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

   9. Taylor expanded in y around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower--.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 67.2

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

   11. Applied rewrites 67.2%

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

Alternative 8: 64.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+65}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+55}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9e+65) (/ y a) (if (<= z 1.8e+55) (/ (fma (- z) y x) t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9e+65) {
		tmp = y / a;
	} else if (z <= 1.8e+55) {
		tmp = fma(-z, y, x) / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9e+65)
		tmp = Float64(y / a);
	elseif (z <= 1.8e+55)
		tmp = Float64(fma(Float64(-z), y, x) / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9e+65], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.8e+55], N[(N[((-z) * y + x), $MachinePrecision] / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9e65 or 1.79999999999999994e55 < z

   1. Initial program 63.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 66.6

         \[\leadsto \color{blue}{\frac{y}{a}} \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\frac{y}{a}} \]

   if -9e65 < z < 1.79999999999999994e55

   1. Initial program 98.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. div-sub N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 67.3

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

   8. Applied rewrites 67.3%

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

      1. lift-*.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lift-neg.f64 N/A

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

      7. lower-fma.f64 67.3

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

   10. Applied rewrites 67.3%

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

Alternative 9: 64.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+65}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9e+65) (/ y a) (if (<= z 1.8e+55) (/ (- x (* y z)) t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9e+65) {
		tmp = y / a;
	} else if (z <= 1.8e+55) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = y / 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 (z <= (-9d+65)) then
        tmp = y / a
    else if (z <= 1.8d+55) then
        tmp = (x - (y * z)) / t
    else
        tmp = y / 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 (z <= -9e+65) {
		tmp = y / a;
	} else if (z <= 1.8e+55) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -9e+65:
		tmp = y / a
	elif z <= 1.8e+55:
		tmp = (x - (y * z)) / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9e+65)
		tmp = Float64(y / a);
	elseif (z <= 1.8e+55)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -9e+65)
		tmp = y / a;
	elseif (z <= 1.8e+55)
		tmp = (x - (y * z)) / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9e+65], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.8e+55], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9e65 or 1.79999999999999994e55 < z

   1. Initial program 63.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 66.6

         \[\leadsto \color{blue}{\frac{y}{a}} \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\frac{y}{a}} \]

   if -9e65 < z < 1.79999999999999994e55

   1. Initial program 98.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

Alternative 10: 65.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+101}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+39}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.7e+101) (/ y a) (if (<= z 2.3e+39) (/ x (- t (* z a))) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.7e+101) {
		tmp = y / a;
	} else if (z <= 2.3e+39) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / 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 (z <= (-1.7d+101)) then
        tmp = y / a
    else if (z <= 2.3d+39) then
        tmp = x / (t - (z * a))
    else
        tmp = y / 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 (z <= -1.7e+101) {
		tmp = y / a;
	} else if (z <= 2.3e+39) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.7e+101:
		tmp = y / a
	elif z <= 2.3e+39:
		tmp = x / (t - (z * a))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.7e+101)
		tmp = Float64(y / a);
	elseif (z <= 2.3e+39)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.7e+101)
		tmp = y / a;
	elseif (z <= 2.3e+39)
		tmp = x / (t - (z * a));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.7e+101], N[(y / a), $MachinePrecision], If[LessEqual[z, 2.3e+39], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.70000000000000009e101 or 2.30000000000000012e39 < z

   1. Initial program 63.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 66.6

         \[\leadsto \color{blue}{\frac{y}{a}} \]

   5. Applied rewrites 66.6%

      \[\leadsto \color{blue}{\frac{y}{a}} \]

   if -1.70000000000000009e101 < z < 2.30000000000000012e39

   1. Initial program 98.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

Alternative 11: 55.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{-47}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.02e-47) (/ y a) (if (<= z 8e+15) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.02e-47) {
		tmp = y / a;
	} else if (z <= 8e+15) {
		tmp = x / t;
	} else {
		tmp = y / 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 (z <= (-1.02d-47)) then
        tmp = y / a
    else if (z <= 8d+15) then
        tmp = x / t
    else
        tmp = y / 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 (z <= -1.02e-47) {
		tmp = y / a;
	} else if (z <= 8e+15) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.02e-47:
		tmp = y / a
	elif z <= 8e+15:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.02e-47)
		tmp = Float64(y / a);
	elseif (z <= 8e+15)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.02e-47)
		tmp = y / a;
	elseif (z <= 8e+15)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.02e-47], N[(y / a), $MachinePrecision], If[LessEqual[z, 8e+15], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.02000000000000002e-47 or 8e15 < z

   1. Initial program 73.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 58.6

         \[\leadsto \color{blue}{\frac{y}{a}} \]

   5. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\frac{y}{a}} \]

   if -1.02000000000000002e-47 < z < 8e15

   1. Initial program 99.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x}{t}} \]
   4. Step-by-step derivation

      1. lower-/.f64 54.8

         \[\leadsto \color{blue}{\frac{x}{t}} \]

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\frac{x}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 35.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{t} \end{array} \]

FPCore

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

C

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

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 / t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x / t), $MachinePrecision]

Derivation

1. Initial program 85.7%

   \[\frac{x - y \cdot z}{t - a \cdot z} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{x}{t}} \]
4. Step-by-step derivation

   1. lower-/.f64 34.1

      \[\leadsto \color{blue}{\frac{x}{t}} \]

5. Applied rewrites 34.1%

   \[\leadsto \color{blue}{\frac{x}{t}} \]
6. Add Preprocessing

Developer Target 1: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x}{t\_1} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))) (t_2 (- (/ x t_1) (/ y (- (/ t z) a)))))
   (if (< z -32113435955957344.0)
     t_2
     (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 t_1)) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x / t_1) - (y / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / t_1);
	} 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 = t - (a * z)
    t_2 = (x / t_1) - (y / ((t / z) - a))
    if (z < (-32113435955957344.0d0)) then
        tmp = t_2
    else if (z < 3.5139522372978296d-86) then
        tmp = (x - (y * z)) * (1.0d0 / t_1)
    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 = t - (a * z);
	double t_2 = (x / t_1) - (y / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (a * z)
	t_2 = (x / t_1) - (y / ((t / z) - a))
	tmp = 0
	if z < -32113435955957344.0:
		tmp = t_2
	elif z < 3.5139522372978296e-86:
		tmp = (x - (y * z)) * (1.0 / t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = Float64(Float64(x / t_1) - Float64(y / Float64(Float64(t / z) - a)))
	tmp = 0.0
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = Float64(Float64(x - Float64(y * z)) * Float64(1.0 / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (a * z);
	t_2 = (x / t_1) - (y / ((t / z) - a));
	tmp = 0.0;
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = (x - (y * z)) * (1.0 / t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / t$95$1), $MachinePrecision] - N[(y / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -32113435955957344.0], t$95$2, If[Less[z, 3.5139522372978296e-86], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2024216 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 4392440296622287/125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))))))

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