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

Percentage Accurate: 85.5% → 96.9%

Time: 9.5s

Alternatives: 11

Speedup: 0.2×

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: 85.5% 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: 96.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{\mathsf{fma}\left(-z, a, t\right)}\right)\\ t_2 := x - z \cdot y\\ t_3 := \frac{t\_2}{t - a \cdot z}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-220}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \frac{a}{z \cdot y - x}, \frac{t}{t\_2}\right)}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

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_3 (/ t_2 (- t (* a z)))))
   (if (<= t_3 -2e-220)
     t_1
     (if (<= t_3 0.0)
       (/ 1.0 (fma z (/ a (- (* z y) x)) (/ t t_2)))
       (if (<= t_3 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);
	double t_3 = t_2 / (t - (a * z));
	double tmp;
	if (t_3 <= -2e-220) {
		tmp = t_1;
	} else if (t_3 <= 0.0) {
		tmp = 1.0 / fma(z, (a / ((z * y) - x)), (t / t_2));
	} else if (t_3 <= ((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(x - Float64(z * y))
	t_3 = Float64(t_2 / Float64(t - Float64(a * z)))
	tmp = 0.0
	if (t_3 <= -2e-220)
		tmp = t_1;
	elseif (t_3 <= 0.0)
		tmp = Float64(1.0 / fma(z, Float64(a / Float64(Float64(z * y) - x)), Float64(t / t_2)));
	elseif (t_3 <= 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[(x - N[(z * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-220], t$95$1, If[LessEqual[t$95$3, 0.0], N[(1.0 / N[(z * N[(a / N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(t / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 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))) < -1.99999999999999998e-220 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 93.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 99.1%

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

   if -1.99999999999999998e-220 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 64.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 64.9

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

      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 64.9

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

   4. Applied rewrites 64.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 94.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 94.8

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

   6. Applied rewrites 94.8%

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

   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 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - z \cdot y}{t - a \cdot z} \leq -2 \cdot 10^{-220}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{\mathsf{fma}\left(-z, a, t\right)}\right)\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \frac{a}{z \cdot y - x}, \frac{t}{x - z \cdot y}\right)}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{\mathsf{fma}\left(-z, a, t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{\mathsf{fma}\left(-z, a, t\right)}\right)\\ t_2 := \frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-298}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \frac{a}{z \cdot y - x}, \frac{t}{x}\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

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 -2e-298)
     t_1
     (if (<= t_2 0.0)
       (/ 1.0 (fma z (/ a (- (* z y) x)) (/ t x)))
       (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 <= -2e-298) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = 1.0 / fma(z, (a / ((z * y) - x)), (t / x));
	} 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 <= -2e-298)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(1.0 / fma(z, Float64(a / Float64(Float64(z * y) - x)), Float64(t / x)));
	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, -2e-298], t$95$1, If[LessEqual[t$95$2, 0.0], N[(1.0 / N[(z * N[(a / N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(t / x), $MachinePrecision]), $MachinePrecision]), $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))) < -1.99999999999999982e-298 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 93.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 98.3%

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

   if -1.99999999999999982e-298 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 52.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 52.6

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

      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 52.6

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

   4. Applied rewrites 52.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 96.1

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 96.1

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

   6. Applied rewrites 96.1%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 82.5

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

   9. Applied rewrites 82.5%

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

   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 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - z \cdot y}{t - a \cdot z} \leq -2 \cdot 10^{-298}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{\mathsf{fma}\left(-z, a, t\right)}\right)\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \frac{a}{z \cdot y - x}, \frac{t}{x}\right)}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{\mathsf{fma}\left(-z, a, t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{\mathsf{fma}\left(-z, a, t\right)}\right)\\ t_2 := \frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-294}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

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 -2e-294)
     t_1
     (if (<= t_2 0.0)
       (/ (- y (/ x z)) a)
       (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 <= -2e-294) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = (y - (x / z)) / a;
	} 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 <= -2e-294)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	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, -2e-294], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $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))) < -2.00000000000000003e-294 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 93.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 98.3%

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

   if -2.00000000000000003e-294 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 54.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 83.0

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

   5. Applied rewrites 83.0%

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

   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 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - z \cdot y}{t - a \cdot z} \leq -2 \cdot 10^{-294}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{\mathsf{fma}\left(-z, a, t\right)}\right)\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{\mathsf{fma}\left(-z, a, t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ t_2 := \frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-294}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+302}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)) (t_2 (/ (- x (* z y)) (- t (* a z)))))
   (if (<= t_2 -2e-294)
     t_2
     (if (<= t_2 0.0) t_1 (if (<= t_2 1e+302) t_2 t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double t_2 = (x - (z * y)) / (t - (a * z));
	double tmp;
	if (t_2 <= -2e-294) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 1e+302) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = (y - (x / z)) / a
    t_2 = (x - (z * y)) / (t - (a * z))
    if (t_2 <= (-2d-294)) then
        tmp = t_2
    else if (t_2 <= 0.0d0) then
        tmp = t_1
    else if (t_2 <= 1d+302) then
        tmp = t_2
    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 t_2 = (x - (z * y)) / (t - (a * z));
	double tmp;
	if (t_2 <= -2e-294) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 1e+302) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	t_2 = (x - (z * y)) / (t - (a * z))
	tmp = 0
	if t_2 <= -2e-294:
		tmp = t_2
	elif t_2 <= 0.0:
		tmp = t_1
	elif t_2 <= 1e+302:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - Float64(x / z)) / a)
	t_2 = Float64(Float64(x - Float64(z * y)) / Float64(t - Float64(a * z)))
	tmp = 0.0
	if (t_2 <= -2e-294)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 1e+302)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - (x / z)) / a;
	t_2 = (x - (z * y)) / (t - (a * z));
	tmp = 0.0;
	if (t_2 <= -2e-294)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 1e+302)
		tmp = t_2;
	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]}, 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, -2e-294], t$95$2, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 1e+302], t$95$2, t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -2.00000000000000003e-294 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.0000000000000001e302

   1. Initial program 97.0%

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

   if -2.00000000000000003e-294 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0 or 1.0000000000000001e302 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 48.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 81.9

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

   5. Applied rewrites 81.9%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - z \cdot y}{t - a \cdot z} \leq -2 \cdot 10^{-294}:\\ \;\;\;\;\frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq 0:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq 10^{+302}:\\ \;\;\;\;\frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -8 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-197}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{+62}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -8e+22)
     t_1
     (if (<= z -1.12e-197)
       (/ x (- t (* a z)))
       (if (<= z 6.4e+62) (/ (- 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 <= -8e+22) {
		tmp = t_1;
	} else if (z <= -1.12e-197) {
		tmp = x / (t - (a * z));
	} else if (z <= 6.4e+62) {
		tmp = (x - (z * y)) / 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 - (x / z)) / a
    if (z <= (-8d+22)) then
        tmp = t_1
    else if (z <= (-1.12d-197)) then
        tmp = x / (t - (a * z))
    else if (z <= 6.4d+62) then
        tmp = (x - (z * y)) / 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 - (x / z)) / a;
	double tmp;
	if (z <= -8e+22) {
		tmp = t_1;
	} else if (z <= -1.12e-197) {
		tmp = x / (t - (a * z));
	} else if (z <= 6.4e+62) {
		tmp = (x - (z * y)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	tmp = 0
	if z <= -8e+22:
		tmp = t_1
	elif z <= -1.12e-197:
		tmp = x / (t - (a * z))
	elif z <= 6.4e+62:
		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 <= -8e+22)
		tmp = t_1;
	elseif (z <= -1.12e-197)
		tmp = Float64(x / Float64(t - Float64(a * z)));
	elseif (z <= 6.4e+62)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	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 <= -8e+22)
		tmp = t_1;
	elseif (z <= -1.12e-197)
		tmp = x / (t - (a * z));
	elseif (z <= 6.4e+62)
		tmp = (x - (z * y)) / t;
	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, -8e+22], t$95$1, If[LessEqual[z, -1.12e-197], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.4e+62], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8e22 or 6.39999999999999968e62 < z

   1. Initial program 68.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 76.6

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

   5. Applied rewrites 76.6%

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

   if -8e22 < z < -1.12e-197

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 7.2

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

   5. Applied rewrites 7.2%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{t - a \cdot z}} \]
   7. 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. lower-*.f64 84.2

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

   8. Applied rewrites 84.2%

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

   if -1.12e-197 < z < 6.39999999999999968e62

   1. Initial program 98.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 79.3

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

   5. Applied rewrites 79.3%

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

Alternative 6: 64.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+114}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-197}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.7e+114)
   (/ y a)
   (if (<= z -1.12e-197)
     (/ x (- t (* a z)))
     (if (<= z 2.2e+68) (/ (- x (* z y)) t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.7e+114) {
		tmp = y / a;
	} else if (z <= -1.12e-197) {
		tmp = x / (t - (a * z));
	} else if (z <= 2.2e+68) {
		tmp = (x - (z * y)) / 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 <= (-1.7d+114)) then
        tmp = y / a
    else if (z <= (-1.12d-197)) then
        tmp = x / (t - (a * z))
    else if (z <= 2.2d+68) then
        tmp = (x - (z * y)) / 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 <= -1.7e+114) {
		tmp = y / a;
	} else if (z <= -1.12e-197) {
		tmp = x / (t - (a * z));
	} else if (z <= 2.2e+68) {
		tmp = (x - (z * y)) / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.7e+114:
		tmp = y / a
	elif z <= -1.12e-197:
		tmp = x / (t - (a * z))
	elif z <= 2.2e+68:
		tmp = (x - (z * y)) / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.7e+114)
		tmp = Float64(y / a);
	elseif (z <= -1.12e-197)
		tmp = Float64(x / Float64(t - Float64(a * z)));
	elseif (z <= 2.2e+68)
		tmp = Float64(Float64(x - Float64(z * y)) / 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 <= -1.7e+114)
		tmp = y / a;
	elseif (z <= -1.12e-197)
		tmp = x / (t - (a * z));
	elseif (z <= 2.2e+68)
		tmp = (x - (z * y)) / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.7e+114], N[(y / a), $MachinePrecision], If[LessEqual[z, -1.12e-197], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.2e+68], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7e114 or 2.19999999999999987e68 < z

   1. Initial program 63.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 59.4

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

   5. Applied rewrites 59.4%

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

   if -1.7e114 < z < -1.12e-197

   1. Initial program 97.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 17.8

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

   5. Applied rewrites 17.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{t - a \cdot z}} \]
   7. 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. lower-*.f64 76.8

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

   8. Applied rewrites 76.8%

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

   if -1.12e-197 < z < 2.19999999999999987e68

   1. Initial program 98.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 78.2

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

   5. Applied rewrites 78.2%

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

Alternative 7: 66.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{z}{a \cdot z - t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -1.35e-65)
   (/ x (fma (- z) a t))
   (if (<= x 9.2e+86) (* (/ z (- (* a z) t)) y) (/ x (- t (* a z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.35e-65) {
		tmp = x / fma(-z, a, t);
	} else if (x <= 9.2e+86) {
		tmp = (z / ((a * z) - t)) * y;
	} else {
		tmp = x / (t - (a * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -1.35e-65)
		tmp = Float64(x / fma(Float64(-z), a, t));
	elseif (x <= 9.2e+86)
		tmp = Float64(Float64(z / Float64(Float64(a * z) - t)) * y);
	else
		tmp = Float64(x / Float64(t - Float64(a * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -1.35e-65], N[(x / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.2e+86], N[(N[(z / N[(N[(a * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3499999999999999e-65

   1. Initial program 89.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 78.6

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

   5. Applied rewrites 78.6%

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

   if -1.3499999999999999e-65 < x < 9.19999999999999958e86

   1. Initial program 86.1%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. associate-*r* N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. mul-1-neg N/A

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

      15. lower-fma.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 62.7

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

   5. Applied rewrites 62.7%

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

   7. Applied rewrites 67.6%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 68.1%

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

   if 9.19999999999999958e86 < x

   1. Initial program 83.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 16.5

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

   5. Applied rewrites 16.5%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{t - a \cdot z}} \]
   7. 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. lower-*.f64 74.5

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

   8. Applied rewrites 74.5%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{z}{a \cdot z - t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+114}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.7e+114)
   (/ y a)
   (if (<= z 2.2e+68) (/ x (fma (- z) a t)) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.7e+114) {
		tmp = y / a;
	} else if (z <= 2.2e+68) {
		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 <= -1.7e+114)
		tmp = Float64(y / a);
	elseif (z <= 2.2e+68)
		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, -1.7e+114], N[(y / a), $MachinePrecision], If[LessEqual[z, 2.2e+68], N[(x / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e114 or 2.19999999999999987e68 < z

   1. Initial program 63.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 59.4

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

   5. Applied rewrites 59.4%

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

   if -1.7e114 < z < 2.19999999999999987e68

   1. Initial program 97.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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%

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

Alternative 9: 65.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+114}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.7e+114) (/ y a) (if (<= z 2.2e+68) (/ x (- t (* a z))) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.7e+114) {
		tmp = y / a;
	} else if (z <= 2.2e+68) {
		tmp = x / (t - (a * z));
	} 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+114)) then
        tmp = y / a
    else if (z <= 2.2d+68) then
        tmp = x / (t - (a * z))
    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+114) {
		tmp = y / a;
	} else if (z <= 2.2e+68) {
		tmp = x / (t - (a * z));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.7e+114:
		tmp = y / a
	elif z <= 2.2e+68:
		tmp = x / (t - (a * z))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.7e+114)
		tmp = Float64(y / a);
	elseif (z <= 2.2e+68)
		tmp = Float64(x / Float64(t - Float64(a * z)));
	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+114)
		tmp = y / a;
	elseif (z <= 2.2e+68)
		tmp = x / (t - (a * z));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.7e+114], N[(y / a), $MachinePrecision], If[LessEqual[z, 2.2e+68], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7e114 or 2.19999999999999987e68 < z

   1. Initial program 63.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 59.4

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

   5. Applied rewrites 59.4%

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

   if -1.7e114 < z < 2.19999999999999987e68

   1. Initial program 97.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 16.6

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

   5. Applied rewrites 16.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{t - a \cdot z}} \]
   7. 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. lower-*.f64 74.9

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

   8. Applied rewrites 74.9%

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

Alternative 10: 54.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8e+22) (/ y a) (if (<= z 1.5e+68) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8e+22) {
		tmp = y / a;
	} else if (z <= 1.5e+68) {
		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 <= (-8d+22)) then
        tmp = y / a
    else if (z <= 1.5d+68) 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 <= -8e+22) {
		tmp = y / a;
	} else if (z <= 1.5e+68) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -8e+22:
		tmp = y / a
	elif z <= 1.5e+68:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8e+22)
		tmp = Float64(y / a);
	elseif (z <= 1.5e+68)
		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 <= -8e+22)
		tmp = y / a;
	elseif (z <= 1.5e+68)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8e+22], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.5e+68], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -8e22 or 1.5000000000000001e68 < z

   1. Initial program 68.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 57.0

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

   5. Applied rewrites 57.0%

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

   if -8e22 < z < 1.5000000000000001e68

   1. Initial program 98.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 58.4

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

   5. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\frac{x}{t}} \]
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 86.8%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 38.8

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

5. Applied rewrites 38.8%

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

Developer Target 1: 97.2% 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 2024268 
(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))))