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

Percentage Accurate: 84.9% → 94.7%

Time: 7.8s

Alternatives: 11

Speedup: 0.7×

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: 94.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{\mathsf{fma}\left(-z, a, t\right)}\right)\\ t_2 := \frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{z \cdot y - x}{a}}{z}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z (fma a z (- t))) y (/ x (fma (- z) a t))))
        (t_2 (/ (- x (* z y)) (- t (* a z)))))
   (if (<= t_2 -5e-324)
     t_1
     (if (<= t_2 0.0)
       (/ (/ (- (* z y) x) a) z)
       (if (<= t_2 INFINITY) t_1 (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / fma(a, z, -t)), y, (x / fma(-z, a, t)));
	double t_2 = (x - (z * y)) / (t - (a * z));
	double tmp;
	if (t_2 <= -5e-324) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = (((z * y) - x) / a) / z;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / fma(a, z, Float64(-t))), y, Float64(x / fma(Float64(-z), a, t)))
	t_2 = Float64(Float64(x - Float64(z * y)) / Float64(t - Float64(a * z)))
	tmp = 0.0
	if (t_2 <= -5e-324)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(z * y) - x) / a) / z);
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision] * y + N[(x / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-324], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / a), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -4.94066e-324 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 93.1%

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

   3. Taylor expanded in y 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. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 97.8%

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

   if -4.94066e-324 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 44.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. neg-sub0 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{0 - \left(t - a \cdot z\right)}}{\mathsf{neg}\left(\left(x - y \cdot z\right)\right)}} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(t - a \cdot z\right)}}{\mathsf{neg}\left(\left(x - y \cdot z\right)\right)}} \]

      8. sub-neg N/A

         \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(t + \left(\mathsf{neg}\left(a \cdot z\right)\right)\right)}}{\mathsf{neg}\left(\left(x - y \cdot z\right)\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot z\right)\right) + t\right)}}{\mathsf{neg}\left(\left(x - y \cdot z\right)\right)}} \]

      10. associate--r+ N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a \cdot z\right)\right)\right) - t}}{\mathsf{neg}\left(\left(x - y \cdot z\right)\right)}} \]

      11. neg-sub0 N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 N/A

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

      14. neg-sub0 N/A

          \[\leadsto \frac{1}{\frac{a \cdot z - t}{\color{blue}{0 - \left(x - y \cdot z\right)}}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot z - t}{0 - \color{blue}{\left(x - y \cdot z\right)}}} \]

      16. sub-neg N/A

          \[\leadsto \frac{1}{\frac{a \cdot z - t}{0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}}} \]

      17. +-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot z - t}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y \cdot z\right)\right) + x\right)}}} \]

      18. associate--r+ N/A

          \[\leadsto \frac{1}{\frac{a \cdot z - t}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - x}}} \]

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. lower--.f64 44.1

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 44.1

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in a around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 87.9

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

   7. Applied rewrites 87.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - z \cdot y}{t - a \cdot z} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{\mathsf{fma}\left(-z, a, t\right)}\right)\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq 0:\\ \;\;\;\;\frac{\frac{z \cdot y - x}{a}}{z}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{\mathsf{fma}\left(-z, a, t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x - z \cdot y}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{y}{t\_1}, \frac{x}{t}\right)\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{z \cdot y - x}{a}}{z}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))) (t_2 (/ (- x (* z y)) t_1)))
   (if (<= t_2 (- INFINITY))
     (fma (- z) (/ y t_1) (/ x t))
     (if (<= t_2 -5e-324)
       t_2
       (if (<= t_2 0.0)
         (/ (/ (- (* z y) x) a) z)
         (if (<= t_2 5e+300) t_2 (/ (- y (/ x z)) a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x - (z * y)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(-z, (y / t_1), (x / t));
	} else if (t_2 <= -5e-324) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (((z * y) - x) / a) / z;
	} else if (t_2 <= 5e+300) {
		tmp = t_2;
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = Float64(Float64(x - Float64(z * y)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(Float64(-z), Float64(y / t_1), Float64(x / t));
	elseif (t_2 <= -5e-324)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(z * y) - x) / a) / z);
	elseif (t_2 <= 5e+300)
		tmp = t_2;
	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[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[((-z) * N[(y / t$95$1), $MachinePrecision] + N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-324], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / a), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$2, 5e+300], t$95$2, 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 59.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -4.94066e-324 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 5.00000000000000026e300

   1. Initial program 99.6%

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

   if -4.94066e-324 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 44.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. neg-sub0 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{0 - \left(t - a \cdot z\right)}}{\mathsf{neg}\left(\left(x - y \cdot z\right)\right)}} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(t - a \cdot z\right)}}{\mathsf{neg}\left(\left(x - y \cdot z\right)\right)}} \]

      8. sub-neg N/A

         \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(t + \left(\mathsf{neg}\left(a \cdot z\right)\right)\right)}}{\mathsf{neg}\left(\left(x - y \cdot z\right)\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot z\right)\right) + t\right)}}{\mathsf{neg}\left(\left(x - y \cdot z\right)\right)}} \]

      10. associate--r+ N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a \cdot z\right)\right)\right) - t}}{\mathsf{neg}\left(\left(x - y \cdot z\right)\right)}} \]

      11. neg-sub0 N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 N/A

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

      14. neg-sub0 N/A

          \[\leadsto \frac{1}{\frac{a \cdot z - t}{\color{blue}{0 - \left(x - y \cdot z\right)}}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot z - t}{0 - \color{blue}{\left(x - y \cdot z\right)}}} \]

      16. sub-neg N/A

          \[\leadsto \frac{1}{\frac{a \cdot z - t}{0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}}} \]

      17. +-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot z - t}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y \cdot z\right)\right) + x\right)}}} \]

      18. associate--r+ N/A

          \[\leadsto \frac{1}{\frac{a \cdot z - t}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - x}}} \]

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. lower--.f64 44.1

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 44.1

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in a around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 87.9

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

   7. Applied rewrites 87.9%

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

   if 5.00000000000000026e300 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 29.7%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. remove-double-neg N/A

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

      13. remove-double-neg N/A

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

      14. neg-mul-1 N/A

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

      15. remove-double-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 83.2

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

   5. Applied rewrites 83.2%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - z \cdot y}{t - a \cdot z} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{y}{t - a \cdot z}, \frac{x}{t}\right)\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq 0:\\ \;\;\;\;\frac{\frac{z \cdot y - x}{a}}{z}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{z \cdot y - x}{a}}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* z y)) (- t (* a z)))))
   (if (<= t_1 -5e-324)
     t_1
     (if (<= t_1 0.0)
       (/ (/ (- (* z y) x) a) z)
       (if (<= t_1 5e+300) t_1 (/ (- y (/ x z)) a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (z * y)) / (t - (a * z));
	double tmp;
	if (t_1 <= -5e-324) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (((z * y) - x) / a) / z;
	} else if (t_1 <= 5e+300) {
		tmp = t_1;
	} else {
		tmp = (y - (x / z)) / 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) :: t_1
    real(8) :: tmp
    t_1 = (x - (z * y)) / (t - (a * z))
    if (t_1 <= (-5d-324)) then
        tmp = t_1
    else if (t_1 <= 0.0d0) then
        tmp = (((z * y) - x) / a) / z
    else if (t_1 <= 5d+300) then
        tmp = t_1
    else
        tmp = (y - (x / z)) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (z * y)) / (t - (a * z));
	double tmp;
	if (t_1 <= -5e-324) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (((z * y) - x) / a) / z;
	} else if (t_1 <= 5e+300) {
		tmp = t_1;
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (z * y)) / (t - (a * z))
	tmp = 0
	if t_1 <= -5e-324:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = (((z * y) - x) / a) / z
	elif t_1 <= 5e+300:
		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(z * y)) / Float64(t - Float64(a * z)))
	tmp = 0.0
	if (t_1 <= -5e-324)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(z * y) - x) / a) / z);
	elseif (t_1 <= 5e+300)
		tmp = t_1;
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - (z * y)) / (t - (a * z));
	tmp = 0.0;
	if (t_1 <= -5e-324)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = (((z * y) - x) / a) / z;
	elseif (t_1 <= 5e+300)
		tmp = t_1;
	else
		tmp = (y - (x / z)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-324], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / a), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5e+300], 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))) < -4.94066e-324 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 5.00000000000000026e300

   1. Initial program 96.8%

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

   if -4.94066e-324 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 44.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. neg-sub0 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{0 - \left(t - a \cdot z\right)}}{\mathsf{neg}\left(\left(x - y \cdot z\right)\right)}} \]

      7. lift--.f64 N/A

         \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(t - a \cdot z\right)}}{\mathsf{neg}\left(\left(x - y \cdot z\right)\right)}} \]

      8. sub-neg N/A

         \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(t + \left(\mathsf{neg}\left(a \cdot z\right)\right)\right)}}{\mathsf{neg}\left(\left(x - y \cdot z\right)\right)}} \]

      9. +-commutative N/A

         \[\leadsto \frac{1}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(a \cdot z\right)\right) + t\right)}}{\mathsf{neg}\left(\left(x - y \cdot z\right)\right)}} \]

      10. associate--r+ N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(a \cdot z\right)\right)\right) - t}}{\mathsf{neg}\left(\left(x - y \cdot z\right)\right)}} \]

      11. neg-sub0 N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 N/A

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

      14. neg-sub0 N/A

          \[\leadsto \frac{1}{\frac{a \cdot z - t}{\color{blue}{0 - \left(x - y \cdot z\right)}}} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1}{\frac{a \cdot z - t}{0 - \color{blue}{\left(x - y \cdot z\right)}}} \]

      16. sub-neg N/A

          \[\leadsto \frac{1}{\frac{a \cdot z - t}{0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y \cdot z\right)\right)\right)}}} \]

      17. +-commutative N/A

          \[\leadsto \frac{1}{\frac{a \cdot z - t}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y \cdot z\right)\right) + x\right)}}} \]

      18. associate--r+ N/A

          \[\leadsto \frac{1}{\frac{a \cdot z - t}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y \cdot z\right)\right)\right) - x}}} \]

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. lower--.f64 44.1

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 44.1

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in a around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 87.9

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

   7. Applied rewrites 87.9%

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

   if 5.00000000000000026e300 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 29.7%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. remove-double-neg N/A

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

      13. remove-double-neg N/A

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

      14. neg-mul-1 N/A

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

      15. remove-double-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 83.2

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

   5. Applied rewrites 83.2%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - z \cdot y}{t - a \cdot z} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq 0:\\ \;\;\;\;\frac{\frac{z \cdot y - x}{a}}{z}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{-93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-175}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{+35}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -2.2e-93)
     t_1
     (if (<= z 2.05e-175)
       (/ x (fma (- z) a t))
       (if (<= z 5.1e+35) (/ (- x (* z y)) 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 (z <= -2.2e-93) {
		tmp = t_1;
	} else if (z <= 2.05e-175) {
		tmp = x / fma(-z, a, t);
	} else if (z <= 5.1e+35) {
		tmp = (x - (z * y)) / 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 (z <= -2.2e-93)
		tmp = t_1;
	elseif (z <= 2.05e-175)
		tmp = Float64(x / fma(Float64(-z), a, t));
	elseif (z <= 5.1e+35)
		tmp = Float64(Float64(x - Float64(z * y)) / 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[z, -2.2e-93], t$95$1, If[LessEqual[z, 2.05e-175], N[(x / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.1e+35], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.19999999999999996e-93 or 5.10000000000000017e35 < z

   1. Initial program 66.4%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. remove-double-neg N/A

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

      13. remove-double-neg N/A

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

      14. neg-mul-1 N/A

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

      15. remove-double-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 74.6

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

   5. Applied rewrites 74.6%

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

   if -2.19999999999999996e-93 < z < 2.04999999999999999e-175

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if 2.04999999999999999e-175 < z < 5.10000000000000017e35

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t}} \]
   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. *-commutative N/A

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

      4. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

Alternative 5: 90.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -2 \cdot 10^{+121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -2e+121)
     t_1
     (if (<= z 1.5e+132) (/ (- x (* z y)) (- t (* a z))) 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 (z <= -2e+121) {
		tmp = t_1;
	} else if (z <= 1.5e+132) {
		tmp = (x - (z * y)) / (t - (a * z));
	} 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 - (x / z)) / a
    if (z <= (-2d+121)) then
        tmp = t_1
    else if (z <= 1.5d+132) then
        tmp = (x - (z * y)) / (t - (a * z))
    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 - (x / z)) / a;
	double tmp;
	if (z <= -2e+121) {
		tmp = t_1;
	} else if (z <= 1.5e+132) {
		tmp = (x - (z * y)) / (t - (a * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	tmp = 0
	if z <= -2e+121:
		tmp = t_1
	elif z <= 1.5e+132:
		tmp = (x - (z * y)) / (t - (a * z))
	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 (z <= -2e+121)
		tmp = t_1;
	elseif (z <= 1.5e+132)
		tmp = Float64(Float64(x - Float64(z * y)) / Float64(t - Float64(a * z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - (x / z)) / a;
	tmp = 0.0;
	if (z <= -2e+121)
		tmp = t_1;
	elseif (z <= 1.5e+132)
		tmp = (x - (z * y)) / (t - (a * z));
	else
		tmp = t_1;
	end
	tmp_2 = 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[z, -2e+121], t$95$1, If[LessEqual[z, 1.5e+132], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.00000000000000007e121 or 1.4999999999999999e132 < z

   1. Initial program 49.2%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. *-rgt-identity N/A

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

      12. remove-double-neg N/A

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

      13. remove-double-neg N/A

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

      14. neg-mul-1 N/A

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

      15. remove-double-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 85.8

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

   5. Applied rewrites 85.8%

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

   if -2.00000000000000007e121 < z < 1.4999999999999999e132

   1. Initial program 94.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+121}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+116}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-175}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.8e+116)
   (/ y a)
   (if (<= z 2.05e-175)
     (/ x (fma (- z) a t))
     (if (<= z 1.5e+72) (/ (- x (* z y)) t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.8e+116) {
		tmp = y / a;
	} else if (z <= 2.05e-175) {
		tmp = x / fma(-z, a, t);
	} else if (z <= 1.5e+72) {
		tmp = (x - (z * y)) / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.8e+116)
		tmp = Float64(y / a);
	elseif (z <= 2.05e-175)
		tmp = Float64(x / fma(Float64(-z), a, t));
	elseif (z <= 1.5e+72)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.8e+116], N[(y / a), $MachinePrecision], If[LessEqual[z, 2.05e-175], N[(x / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.5e+72], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.80000000000000004e116 or 1.50000000000000001e72 < z

   1. Initial program 53.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 64.9

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

   5. Applied rewrites 64.9%

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

   if -2.80000000000000004e116 < z < 2.04999999999999999e-175

   1. Initial program 97.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 74.9

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

   5. Applied rewrites 74.9%

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

   if 2.04999999999999999e-175 < z < 1.50000000000000001e72

   1. Initial program 99.7%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t}} \]
   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. *-commutative N/A

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

      4. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

Alternative 7: 54.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+70}:\\ \;\;\;\;\frac{\left(-y\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.2e-93)
   (/ y a)
   (if (<= z 1.9e-31) (/ x t) (if (<= z 7.5e+70) (/ (* (- y) z) t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.2e-93) {
		tmp = y / a;
	} else if (z <= 1.9e-31) {
		tmp = x / t;
	} else if (z <= 7.5e+70) {
		tmp = (-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 <= (-2.2d-93)) then
        tmp = y / a
    else if (z <= 1.9d-31) then
        tmp = x / t
    else if (z <= 7.5d+70) then
        tmp = (-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 <= -2.2e-93) {
		tmp = y / a;
	} else if (z <= 1.9e-31) {
		tmp = x / t;
	} else if (z <= 7.5e+70) {
		tmp = (-y * z) / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.2e-93:
		tmp = y / a
	elif z <= 1.9e-31:
		tmp = x / t
	elif z <= 7.5e+70:
		tmp = (-y * z) / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.2e-93)
		tmp = Float64(y / a);
	elseif (z <= 1.9e-31)
		tmp = Float64(x / t);
	elseif (z <= 7.5e+70)
		tmp = Float64(Float64(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 <= -2.2e-93)
		tmp = y / a;
	elseif (z <= 1.9e-31)
		tmp = x / t;
	elseif (z <= 7.5e+70)
		tmp = (-y * z) / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.2e-93], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.9e-31], N[(x / t), $MachinePrecision], If[LessEqual[z, 7.5e+70], N[(N[((-y) * z), $MachinePrecision] / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.19999999999999996e-93 or 7.50000000000000031e70 < z

   1. Initial program 64.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 55.1

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

   5. Applied rewrites 55.1%

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

   if -2.19999999999999996e-93 < z < 1.9e-31

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 66.3

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

   5. Applied rewrites 66.3%

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

   if 1.9e-31 < z < 7.50000000000000031e70

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{x - y \cdot z}{t}} \]
   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. *-commutative N/A

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

      4. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 54.6%

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

Alternative 8: 66.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.9 \cdot 10^{-112}:\\ \;\;\;\;\frac{z}{a \cdot z - t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (fma (- z) a t))))
   (if (<= x -6.5e+42)
     t_1
     (if (<= x 7.9e-112) (* (/ z (- (* a z) t)) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / fma(-z, a, t);
	double tmp;
	if (x <= -6.5e+42) {
		tmp = t_1;
	} else if (x <= 7.9e-112) {
		tmp = (z / ((a * z) - t)) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / fma(Float64(-z), a, t))
	tmp = 0.0
	if (x <= -6.5e+42)
		tmp = t_1;
	elseif (x <= 7.9e-112)
		tmp = Float64(Float64(z / Float64(Float64(a * z) - t)) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.5e+42], t$95$1, If[LessEqual[x, 7.9e-112], N[(N[(z / N[(N[(a * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.50000000000000052e42 or 7.8999999999999999e-112 < x

   1. Initial program 85.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 72.7

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

   5. Applied rewrites 72.7%

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

   if -6.50000000000000052e42 < x < 7.8999999999999999e-112

   1. Initial program 78.9%

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

   3. Taylor expanded in y 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. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 68.7%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{elif}\;x \leq 7.9 \cdot 10^{-112}:\\ \;\;\;\;\frac{z}{a \cdot z - t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+116}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+72}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.8e+116)
   (/ y a)
   (if (<= z 1.6e+72) (/ x (fma (- z) a t)) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.8e+116) {
		tmp = y / a;
	} else if (z <= 1.6e+72) {
		tmp = x / fma(-z, a, t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.8e+116)
		tmp = Float64(y / a);
	elseif (z <= 1.6e+72)
		tmp = Float64(x / fma(Float64(-z), a, t));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.8e+116], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.6e+72], N[(x / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.80000000000000004e116 or 1.6000000000000001e72 < z

   1. Initial program 53.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 64.9

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

   5. Applied rewrites 64.9%

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

   if -2.80000000000000004e116 < z < 1.6000000000000001e72

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 69.7

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

   5. Applied rewrites 69.7%

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

Alternative 10: 53.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+70}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.2e-93) (/ y a) (if (<= z 5.6e+70) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.2e-93) {
		tmp = y / a;
	} else if (z <= 5.6e+70) {
		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 <= (-2.2d-93)) then
        tmp = y / a
    else if (z <= 5.6d+70) 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 <= -2.2e-93) {
		tmp = y / a;
	} else if (z <= 5.6e+70) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.2e-93:
		tmp = y / a
	elif z <= 5.6e+70:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.2e-93)
		tmp = Float64(y / a);
	elseif (z <= 5.6e+70)
		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 <= -2.2e-93)
		tmp = y / a;
	elseif (z <= 5.6e+70)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.2e-93], N[(y / a), $MachinePrecision], If[LessEqual[z, 5.6e+70], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.19999999999999996e-93 or 5.59999999999999979e70 < z

   1. Initial program 64.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 55.1

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

   5. Applied rewrites 55.1%

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

   if -2.19999999999999996e-93 < z < 5.59999999999999979e70

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 59.9

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

   5. Applied rewrites 59.9%

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

Alternative 11: 35.2% 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 82.5%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 34.6

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

5. Applied rewrites 34.6%

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

Developer Target 1: 97.4% 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 2024273 
(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))))