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

Percentage Accurate: 85.3% → 96.6%

Time: 7.3s

Alternatives: 12

Speedup: 0.8×

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: 85.3% 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: 96.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, z, -t\right)\\ t_2 := \frac{z}{t\_1}\\ t_3 := x - z \cdot y\\ t_4 := t - a \cdot z\\ t_5 := \frac{t\_3}{t\_4}\\ \mathbf{if}\;t\_5 \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, y, \frac{x}{t\_4}\right)\\ \mathbf{elif}\;t\_5 \leq 10^{-64}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \frac{a}{z \cdot y - x}, \frac{t}{t\_3}\right)}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_2 \cdot y - \frac{x}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a z (- t)))
        (t_2 (/ z t_1))
        (t_3 (- x (* z y)))
        (t_4 (- t (* a z)))
        (t_5 (/ t_3 t_4)))
   (if (<= t_5 -5e-116)
     (fma t_2 y (/ x t_4))
     (if (<= t_5 1e-64)
       (/ 1.0 (fma z (/ a (- (* z y) x)) (/ t t_3)))
       (if (<= t_5 INFINITY) (- (* t_2 y) (/ x t_1)) (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, z, -t);
	double t_2 = z / t_1;
	double t_3 = x - (z * y);
	double t_4 = t - (a * z);
	double t_5 = t_3 / t_4;
	double tmp;
	if (t_5 <= -5e-116) {
		tmp = fma(t_2, y, (x / t_4));
	} else if (t_5 <= 1e-64) {
		tmp = 1.0 / fma(z, (a / ((z * y) - x)), (t / t_3));
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (t_2 * y) - (x / t_1);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, z, Float64(-t))
	t_2 = Float64(z / t_1)
	t_3 = Float64(x - Float64(z * y))
	t_4 = Float64(t - Float64(a * z))
	t_5 = Float64(t_3 / t_4)
	tmp = 0.0
	if (t_5 <= -5e-116)
		tmp = fma(t_2, y, Float64(x / t_4));
	elseif (t_5 <= 1e-64)
		tmp = Float64(1.0 / fma(z, Float64(a / Float64(Float64(z * y) - x)), Float64(t / t_3)));
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(t_2 * y) - Float64(x / t_1));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * z + (-t)), $MachinePrecision]}, Block[{t$95$2 = N[(z / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, -5e-116], N[(t$95$2 * y + N[(x / t$95$4), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 1e-64], N[(1.0 / N[(z * N[(a / N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] + N[(t / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[(t$95$2 * y), $MachinePrecision] - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -5.0000000000000003e-116

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

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

      1. *-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 99.8%

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

   if -5.0000000000000003e-116 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 9.99999999999999965e-65

   1. Initial program 87.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 86.0

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

      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 86.0

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

   4. Applied rewrites 86.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot z - t}{z \cdot y - x}}} \]
   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. lower-neg.f64 N/A

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

      11. lower-/.f64 92.8

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

   6. Applied rewrites 92.8%

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

   if 9.99999999999999965e-65 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 90.4%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. distribute-neg-frac N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 41.5%

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

   6. Applied rewrites 99.7%

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

   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 4 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - z \cdot y}{t - a \cdot z} \leq -5 \cdot 10^{-116}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{t - a \cdot z}\right)\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq 10^{-64}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(z, \frac{a}{z \cdot y - x}, \frac{t}{x - z \cdot y}\right)}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq \infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(a, z, -t\right)} \cdot y - \frac{x}{\mathsf{fma}\left(a, z, -t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, z, -t\right)\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+169}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-82}:\\ \;\;\;\;\frac{z \cdot y}{a \cdot z - t} + \frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t\_1} \cdot y - \frac{x}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a z (- t))))
   (if (<= z -3.2e+169)
     (/ (- y (/ x z)) a)
     (if (<= z 6e-82)
       (+ (/ (* z y) (- (* a z) t)) (/ x (- t (* a z))))
       (- (* (/ z t_1) y) (/ x t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, z, -t);
	double tmp;
	if (z <= -3.2e+169) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 6e-82) {
		tmp = ((z * y) / ((a * z) - t)) + (x / (t - (a * z)));
	} else {
		tmp = ((z / t_1) * y) - (x / t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, z, Float64(-t))
	tmp = 0.0
	if (z <= -3.2e+169)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (z <= 6e-82)
		tmp = Float64(Float64(Float64(z * y) / Float64(Float64(a * z) - t)) + Float64(x / Float64(t - Float64(a * z))));
	else
		tmp = Float64(Float64(Float64(z / t_1) * y) - Float64(x / t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * z + (-t)), $MachinePrecision]}, If[LessEqual[z, -3.2e+169], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, 6e-82], N[(N[(N[(z * y), $MachinePrecision] / N[(N[(a * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z / t$95$1), $MachinePrecision] * y), $MachinePrecision] - N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.1999999999999998e169

   1. Initial program 57.3%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x - y \cdot z}{a \cdot z}} \]
   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. neg-mul-1 N/A

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

      13. remove-double-neg N/A

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

      14. +-commutative N/A

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

      15. mul-1-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -3.1999999999999998e169 < z < 5.9999999999999998e-82

   1. Initial program 97.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. lower-+.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. neg-sub0 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. associate--r+ N/A

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

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 97.2

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

   4. Applied rewrites 97.2%

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

   if 5.9999999999999998e-82 < z

   1. Initial program 83.3%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. flip-- N/A

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

      5. distribute-neg-frac N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 37.9%

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

   6. Applied rewrites 92.7%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+169}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-82}:\\ \;\;\;\;\frac{z \cdot y}{a \cdot z - t} + \frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(a, z, -t\right)} \cdot y - \frac{x}{\mathsf{fma}\left(a, z, -t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - a \cdot z}\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+169}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+36}:\\ \;\;\;\;\frac{z \cdot y}{a \cdot z - t} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* a z)))))
   (if (<= z -3.2e+169)
     (/ (- y (/ x z)) a)
     (if (<= z 5e+36)
       (+ (/ (* z y) (- (* a z) t)) t_1)
       (fma (/ z (fma a z (- t))) y t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (a * z));
	double tmp;
	if (z <= -3.2e+169) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 5e+36) {
		tmp = ((z * y) / ((a * z) - t)) + t_1;
	} else {
		tmp = fma((z / fma(a, z, -t)), y, t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(t - Float64(a * z)))
	tmp = 0.0
	if (z <= -3.2e+169)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (z <= 5e+36)
		tmp = Float64(Float64(Float64(z * y) / Float64(Float64(a * z) - t)) + t_1);
	else
		tmp = fma(Float64(z / fma(a, z, Float64(-t))), y, t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.2e+169], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, 5e+36], N[(N[(N[(z * y), $MachinePrecision] / N[(N[(a * z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(z / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision] * y + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.1999999999999998e169

   1. Initial program 57.3%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x - y \cdot z}{a \cdot z}} \]
   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. neg-mul-1 N/A

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

      13. remove-double-neg N/A

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

      14. +-commutative N/A

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

      15. mul-1-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -3.1999999999999998e169 < z < 4.99999999999999977e36

   1. Initial program 97.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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. 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. lower-+.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. neg-sub0 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. associate--r+ N/A

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

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 N/A

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

      20. lower-/.f64 97.6

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

   4. Applied rewrites 97.6%

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

   if 4.99999999999999977e36 < z

   1. Initial program 75.3%

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

   3. Taylor expanded in x around 0

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

      1. *-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 89.2%

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

Alternative 4: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+169}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-82}:\\ \;\;\;\;\frac{x - z \cdot y}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{t\_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))))
   (if (<= z -3.2e+169)
     (/ (- y (/ x z)) a)
     (if (<= z 6e-82)
       (/ (- x (* z y)) t_1)
       (fma (/ z (fma a z (- t))) y (/ x t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double tmp;
	if (z <= -3.2e+169) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 6e-82) {
		tmp = (x - (z * y)) / t_1;
	} else {
		tmp = fma((z / fma(a, z, -t)), y, (x / t_1));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	tmp = 0.0
	if (z <= -3.2e+169)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (z <= 6e-82)
		tmp = Float64(Float64(x - Float64(z * y)) / t_1);
	else
		tmp = fma(Float64(z / fma(a, z, Float64(-t))), y, Float64(x / t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.2e+169], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, 6e-82], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(z / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision] * y + N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.1999999999999998e169

   1. Initial program 57.3%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x - y \cdot z}{a \cdot z}} \]
   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. neg-mul-1 N/A

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

      13. remove-double-neg N/A

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

      14. +-commutative N/A

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

      15. mul-1-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -3.1999999999999998e169 < z < 5.9999999999999998e-82

   1. Initial program 97.2%

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

   if 5.9999999999999998e-82 < z

   1. Initial program 83.3%

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

   3. Taylor expanded in x around 0

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

      1. *-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 92.7%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+169}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-82}:\\ \;\;\;\;\frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{t - a \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - a \cdot z}\\ \mathbf{if}\;a \leq -1.45 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{+31}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{+175}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* a z)))))
   (if (<= a -1.45e-30)
     t_1
     (if (<= a 1.4e+31)
       (/ (- x (* z y)) t)
       (if (<= a 2.5e+175) t_1 (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (a * z));
	double tmp;
	if (a <= -1.45e-30) {
		tmp = t_1;
	} else if (a <= 1.4e+31) {
		tmp = (x - (z * y)) / t;
	} else if (a <= 2.5e+175) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = x / (t - (a * z))
    if (a <= (-1.45d-30)) then
        tmp = t_1
    else if (a <= 1.4d+31) then
        tmp = (x - (z * y)) / t
    else if (a <= 2.5d+175) then
        tmp = t_1
    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 t_1 = x / (t - (a * z));
	double tmp;
	if (a <= -1.45e-30) {
		tmp = t_1;
	} else if (a <= 1.4e+31) {
		tmp = (x - (z * y)) / t;
	} else if (a <= 2.5e+175) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x / (t - (a * z))
	tmp = 0
	if a <= -1.45e-30:
		tmp = t_1
	elif a <= 1.4e+31:
		tmp = (x - (z * y)) / t
	elif a <= 2.5e+175:
		tmp = t_1
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(t - Float64(a * z)))
	tmp = 0.0
	if (a <= -1.45e-30)
		tmp = t_1;
	elseif (a <= 1.4e+31)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (a <= 2.5e+175)
		tmp = t_1;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x / (t - (a * z));
	tmp = 0.0;
	if (a <= -1.45e-30)
		tmp = t_1;
	elseif (a <= 1.4e+31)
		tmp = (x - (z * y)) / t;
	elseif (a <= 2.5e+175)
		tmp = t_1;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.45e-30], t$95$1, If[LessEqual[a, 1.4e+31], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[a, 2.5e+175], t$95$1, N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.44999999999999995e-30 or 1.40000000000000008e31 < a < 2.5e175

   1. Initial program 88.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 65.6

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

   5. Applied rewrites 65.6%

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

   if -1.44999999999999995e-30 < a < 1.40000000000000008e31

   1. Initial program 95.4%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if 2.5e175 < a

   1. Initial program 61.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 62.4

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

   5. Applied rewrites 62.4%

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

Alternative 6: 69.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{+47}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-z, y, x\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.4e+47)
   (/ (- x (* z y)) t)
   (if (<= t 7.2e-10) (/ (- y (/ x z)) a) (/ (fma (- z) y x) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.4e+47) {
		tmp = (x - (z * y)) / t;
	} else if (t <= 7.2e-10) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = fma(-z, y, x) / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.4e+47)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (t <= 7.2e-10)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(fma(Float64(-z), y, x) / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.4e+47], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 7.2e-10], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[((-z) * y + x), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.39999999999999994e47

   1. Initial program 89.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   if -1.39999999999999994e47 < t < 7.2e-10

   1. Initial program 90.2%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x - y \cdot z}{a \cdot z}} \]
   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. neg-mul-1 N/A

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

      13. remove-double-neg N/A

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

      14. +-commutative N/A

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

      15. mul-1-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 72.6

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

   5. Applied rewrites 72.6%

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

   if 7.2e-10 < t

   1. Initial program 86.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 76.0

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

   5. Applied rewrites 76.0%

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

   7. Applied rewrites 76.0%

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

Alternative 7: 65.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(a, z, -t\right)} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.2e+97)
   (/ y a)
   (if (<= z 3.4e-23) (/ x (- t (* a z))) (* (/ z (fma a z (- t))) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.2e+97) {
		tmp = y / a;
	} else if (z <= 3.4e-23) {
		tmp = x / (t - (a * z));
	} else {
		tmp = (z / fma(a, z, -t)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.2e+97)
		tmp = Float64(y / a);
	elseif (z <= 3.4e-23)
		tmp = Float64(x / Float64(t - Float64(a * z)));
	else
		tmp = Float64(Float64(z / fma(a, z, Float64(-t))) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.2e+97], N[(y / a), $MachinePrecision], If[LessEqual[z, 3.4e-23], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.19999999999999977e97

   1. Initial program 65.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

   if -8.19999999999999977e97 < z < 3.4000000000000001e-23

   1. Initial program 97.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

   if 3.4000000000000001e-23 < z

   1. Initial program 79.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{t - a \cdot z}} \]
   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 57.8

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

   5. Applied rewrites 57.8%

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

   7. Applied rewrites 69.4%

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

Alternative 8: 64.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(a, z, -t\right)} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.2e+97)
   (/ y a)
   (if (<= z 3.2e-23) (/ x (- t (* a z))) (* (/ y (fma a z (- t))) z))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.2e+97) {
		tmp = y / a;
	} else if (z <= 3.2e-23) {
		tmp = x / (t - (a * z));
	} else {
		tmp = (y / fma(a, z, -t)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.2e+97)
		tmp = Float64(y / a);
	elseif (z <= 3.2e-23)
		tmp = Float64(x / Float64(t - Float64(a * z)));
	else
		tmp = Float64(Float64(y / fma(a, z, Float64(-t))) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.2e+97], N[(y / a), $MachinePrecision], If[LessEqual[z, 3.2e-23], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.19999999999999977e97

   1. Initial program 65.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

   if -8.19999999999999977e97 < z < 3.19999999999999976e-23

   1. Initial program 97.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

   if 3.19999999999999976e-23 < z

   1. Initial program 79.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{t - a \cdot z}} \]
   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 57.8

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

   5. Applied rewrites 57.8%

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

   7. Applied rewrites 65.6%

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

Alternative 9: 88.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+169}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - a \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.2e+169) (/ (- y (/ x z)) a) (/ (- x (* z y)) (- t (* a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.2e+169) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (z * y)) / (t - (a * z));
	}
	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 <= (-3.2d+169)) then
        tmp = (y - (x / z)) / a
    else
        tmp = (x - (z * y)) / (t - (a * z))
    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 <= -3.2e+169) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (z * y)) / (t - (a * z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.2e+169:
		tmp = (y - (x / z)) / a
	else:
		tmp = (x - (z * y)) / (t - (a * z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.2e+169)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(Float64(x - Float64(z * y)) / Float64(t - Float64(a * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.2e+169)
		tmp = (y - (x / z)) / a;
	else
		tmp = (x - (z * y)) / (t - (a * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.2e+169], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1999999999999998e169

   1. Initial program 57.3%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x - y \cdot z}{a \cdot z}} \]
   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. neg-mul-1 N/A

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

      13. remove-double-neg N/A

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

      14. +-commutative N/A

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

      15. mul-1-neg N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -3.1999999999999998e169 < z

   1. Initial program 92.4%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+169}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - a \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.2e+97) (/ y a) (if (<= z 5.5e-23) (/ x (- t (* a z))) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.2e+97) {
		tmp = y / a;
	} else if (z <= 5.5e-23) {
		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 <= (-8.2d+97)) then
        tmp = y / a
    else if (z <= 5.5d-23) 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 <= -8.2e+97) {
		tmp = y / a;
	} else if (z <= 5.5e-23) {
		tmp = x / (t - (a * z));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -8.2e+97:
		tmp = y / a
	elif z <= 5.5e-23:
		tmp = x / (t - (a * z))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.2e+97)
		tmp = Float64(y / a);
	elseif (z <= 5.5e-23)
		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 <= -8.2e+97)
		tmp = y / a;
	elseif (z <= 5.5e-23)
		tmp = x / (t - (a * z));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.2e+97], N[(y / a), $MachinePrecision], If[LessEqual[z, 5.5e-23], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.19999999999999977e97 or 5.5000000000000001e-23 < z

   1. Initial program 74.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 56.9

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

   5. Applied rewrites 56.9%

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

   if -8.19999999999999977e97 < z < 5.5000000000000001e-23

   1. Initial program 97.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

Alternative 11: 54.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+47}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.05e+47) (/ y a) (if (<= z 1.3e-31) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.05e+47) {
		tmp = y / a;
	} else if (z <= 1.3e-31) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1.05d+47)) then
        tmp = y / a
    else if (z <= 1.3d-31) then
        tmp = x / t
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.05e+47) {
		tmp = y / a;
	} else if (z <= 1.3e-31) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.05e+47:
		tmp = y / a
	elif z <= 1.3e-31:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.05e+47)
		tmp = Float64(y / a);
	elseif (z <= 1.3e-31)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.05e+47)
		tmp = y / a;
	elseif (z <= 1.3e-31)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.05e+47], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.3e-31], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05e47 or 1.29999999999999998e-31 < z

   1. Initial program 76.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 53.0

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

   5. Applied rewrites 53.0%

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

   if -1.05e47 < z < 1.29999999999999998e-31

   1. Initial program 99.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 51.3

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

   5. Applied rewrites 51.3%

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

Alternative 12: 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 88.9%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 34.2

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

5. Applied rewrites 34.2%

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

Developer Target 1: 97.6% 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 2024295 
(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))))