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

Percentage Accurate: 85.3% → 98.7%

Time: 10.1s

Alternatives: 14

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{t\_1}\right)\\ t_3 := \frac{x - z \cdot y}{t\_1}\\ t_4 := \mathsf{fma}\left(y, z, -x\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-40}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-55}:\\ \;\;\;\;\frac{-1}{\frac{t}{t\_4} - \frac{z}{t\_4} \cdot a}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z)))
        (t_2 (fma (/ z (fma a z (- t))) y (/ x t_1)))
        (t_3 (/ (- x (* z y)) t_1))
        (t_4 (fma y z (- x))))
   (if (<= t_3 -2e-40)
     t_2
     (if (<= t_3 2e-55)
       (/ -1.0 (- (/ t t_4) (* (/ z t_4) a)))
       (if (<= t_3 INFINITY) t_2 (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = fma((z / fma(a, z, -t)), y, (x / t_1));
	double t_3 = (x - (z * y)) / t_1;
	double t_4 = fma(y, z, -x);
	double tmp;
	if (t_3 <= -2e-40) {
		tmp = t_2;
	} else if (t_3 <= 2e-55) {
		tmp = -1.0 / ((t / t_4) - ((z / t_4) * a));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = fma(Float64(z / fma(a, z, Float64(-t))), y, Float64(x / t_1))
	t_3 = Float64(Float64(x - Float64(z * y)) / t_1)
	t_4 = fma(y, z, Float64(-x))
	tmp = 0.0
	if (t_3 <= -2e-40)
		tmp = t_2;
	elseif (t_3 <= 2e-55)
		tmp = Float64(-1.0 / Float64(Float64(t / t_4) - Float64(Float64(z / t_4) * a)));
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision] * y + N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(y * z + (-x)), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-40], t$95$2, If[LessEqual[t$95$3, 2e-55], N[(-1.0 / N[(N[(t / t$95$4), $MachinePrecision] - N[(N[(z / t$95$4), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$2, N[(y / a), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.9999999999999999e-40 or 1.99999999999999999e-55 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 87.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 -1.9999999999999999e-40 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.99999999999999999e-55

   1. Initial program 84.9%

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

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} - {\left(y \cdot z\right)}^{3}}{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}} \cdot \frac{1}{t - a \cdot z} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}}}} \cdot \frac{1}{t - a \cdot z} \]

      6. frac-2neg N/A

         \[\leadsto \frac{1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}}} \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(t - a \cdot z\right)\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}}} \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(t - a \cdot z\right)\right)} \]

      8. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot -1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(t - a \cdot z\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(t - a \cdot z\right)\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(t - a \cdot z\right)\right)\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(t - a \cdot z\right)\right)\right)}} \]

   4. Applied rewrites 83.7%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. sub-neg N/A

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

      15. mul-1-neg N/A

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

      16. lower-fma.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 98.5

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

   7. Applied rewrites 98.5%

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

   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 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - z \cdot y}{t - a \cdot z} \leq -2 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{t - a \cdot z}\right)\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq 2 \cdot 10^{-55}:\\ \;\;\;\;\frac{-1}{\frac{t}{\mathsf{fma}\left(y, z, -x\right)} - \frac{z}{\mathsf{fma}\left(y, z, -x\right)} \cdot a}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{t - a \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - z \cdot y\\ t_2 := \frac{t\_1}{t - a \cdot z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, z, -x\right)}{a}}{z}\\ \mathbf{elif}\;t\_2 \leq 10^{+267}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* z y))) (t_2 (/ t_1 (- t (* a z)))))
   (if (<= t_2 -1e-309)
     t_2
     (if (<= t_2 0.0)
       (/ (/ (fma y z (- x)) a) z)
       (if (<= t_2 1e+267)
         (/ t_1 (fma (- z) a t))
         (if (<= t_2 INFINITY)
           (fma (/ z (fma a z (- t))) y (/ x t))
           (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (z * y);
	double t_2 = t_1 / (t - (a * z));
	double tmp;
	if (t_2 <= -1e-309) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (fma(y, z, -x) / a) / z;
	} else if (t_2 <= 1e+267) {
		tmp = t_1 / fma(-z, a, t);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma((z / fma(a, z, -t)), y, (x / t));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(z * y))
	t_2 = Float64(t_1 / Float64(t - Float64(a * z)))
	tmp = 0.0
	if (t_2 <= -1e-309)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(fma(y, z, Float64(-x)) / a) / z);
	elseif (t_2 <= 1e+267)
		tmp = Float64(t_1 / fma(Float64(-z), a, t));
	elseif (t_2 <= Inf)
		tmp = fma(Float64(z / fma(a, z, Float64(-t))), y, Float64(x / t));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-309], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(N[(y * z + (-x)), $MachinePrecision] / a), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$2, 1e+267], N[(t$95$1 / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(z / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision] * y + N[(x / t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.000000000000002e-309

   1. Initial program 94.3%

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

   if -1.000000000000002e-309 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 45.4%

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

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} - {\left(y \cdot z\right)}^{3}}{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}} \cdot \frac{1}{t - a \cdot z} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}}}} \cdot \frac{1}{t - a \cdot z} \]

      6. frac-2neg N/A

         \[\leadsto \frac{1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}}} \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(t - a \cdot z\right)\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}}} \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(t - a \cdot z\right)\right)} \]

      8. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot -1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(t - a \cdot z\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(t - a \cdot z\right)\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(t - a \cdot z\right)\right)\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(t - a \cdot z\right)\right)\right)}} \]

   4. Applied rewrites 45.4%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 16.0

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

   7. Applied rewrites 16.0%

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

   9. Applied rewrites 77.5%

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

   if -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 9.9999999999999997e266

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if 9.9999999999999997e266 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 45.9%

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

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 96.7%

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

   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 5 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - z \cdot y}{t - a \cdot z} \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, z, -x\right)}{a}}{z}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq 10^{+267}:\\ \;\;\;\;\frac{x - z \cdot y}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - z \cdot y\\ t_2 := \frac{t\_1}{t - a \cdot z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, z, -x\right)}{a}}{z}\\ \mathbf{elif}\;t\_2 \leq 10^{+295}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(a, z, -t\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* z y))) (t_2 (/ t_1 (- t (* a z)))))
   (if (<= t_2 -1e-309)
     t_2
     (if (<= t_2 0.0)
       (/ (/ (fma y z (- x)) a) z)
       (if (<= t_2 1e+295)
         (/ t_1 (fma (- z) a t))
         (if (<= t_2 INFINITY) (* (/ z (fma a z (- t))) y) (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (z * y);
	double t_2 = t_1 / (t - (a * z));
	double tmp;
	if (t_2 <= -1e-309) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (fma(y, z, -x) / a) / z;
	} else if (t_2 <= 1e+295) {
		tmp = t_1 / fma(-z, a, t);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (z / fma(a, z, -t)) * y;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(z * y))
	t_2 = Float64(t_1 / Float64(t - Float64(a * z)))
	tmp = 0.0
	if (t_2 <= -1e-309)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(fma(y, z, Float64(-x)) / a) / z);
	elseif (t_2 <= 1e+295)
		tmp = Float64(t_1 / fma(Float64(-z), a, t));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(z / fma(a, z, Float64(-t))) * y);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-309], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(N[(y * z + (-x)), $MachinePrecision] / a), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$2, 1e+295], N[(t$95$1 / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(z / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.000000000000002e-309

   1. Initial program 94.3%

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

   if -1.000000000000002e-309 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 45.4%

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

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} - {\left(y \cdot z\right)}^{3}}{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}} \cdot \frac{1}{t - a \cdot z} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}}}} \cdot \frac{1}{t - a \cdot z} \]

      6. frac-2neg N/A

         \[\leadsto \frac{1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}}} \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(t - a \cdot z\right)\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}}} \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(t - a \cdot z\right)\right)} \]

      8. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot -1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(t - a \cdot z\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(t - a \cdot z\right)\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(t - a \cdot z\right)\right)\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(t - a \cdot z\right)\right)\right)}} \]

   4. Applied rewrites 45.4%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 16.0

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

   7. Applied rewrites 16.0%

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

   9. Applied rewrites 77.5%

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

   if -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 9.9999999999999998e294

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if 9.9999999999999998e294 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 43.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 13.8

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

   5. Applied rewrites 13.8%

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

   7. Applied rewrites 70.1%

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

   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 5 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - z \cdot y}{t - a \cdot z} \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, z, -x\right)}{a}}{z}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq 10^{+295}:\\ \;\;\;\;\frac{x - z \cdot y}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq \infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(a, z, -t\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{t\_1}\right)\\ t_3 := \frac{x - z \cdot y}{t\_1}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, z, -x\right)}{a}}{z}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z)))
        (t_2 (fma (/ z (fma a z (- t))) y (/ x t_1)))
        (t_3 (/ (- x (* z y)) t_1)))
   (if (<= t_3 -1e-309)
     t_2
     (if (<= t_3 0.0)
       (/ (/ (fma y z (- x)) a) z)
       (if (<= t_3 INFINITY) t_2 (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = fma((z / fma(a, z, -t)), y, (x / t_1));
	double t_3 = (x - (z * y)) / t_1;
	double tmp;
	if (t_3 <= -1e-309) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = (fma(y, z, -x) / a) / z;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = fma(Float64(z / fma(a, z, Float64(-t))), y, Float64(x / t_1))
	t_3 = Float64(Float64(x - Float64(z * y)) / t_1)
	tmp = 0.0
	if (t_3 <= -1e-309)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(fma(y, z, Float64(-x)) / a) / z);
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision] * y + N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-309], t$95$2, If[LessEqual[t$95$3, 0.0], N[(N[(N[(y * z + (-x)), $MachinePrecision] / a), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$2, N[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.000000000000002e-309 or -0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 91.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 97.4%

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

   if -1.000000000000002e-309 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -0.0

   1. Initial program 45.4%

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

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

      3. lift--.f64 N/A

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

      4. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} - {\left(y \cdot z\right)}^{3}}{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}} \cdot \frac{1}{t - a \cdot z} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}}}} \cdot \frac{1}{t - a \cdot z} \]

      6. frac-2neg N/A

         \[\leadsto \frac{1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}}} \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(t - a \cdot z\right)\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}}} \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\left(t - a \cdot z\right)\right)} \]

      8. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot -1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(t - a \cdot z\right)\right)\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(t - a \cdot z\right)\right)\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(t - a \cdot z\right)\right)\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{-1}{\color{blue}{\frac{x \cdot x + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) + x \cdot \left(y \cdot z\right)\right)}{{x}^{3} - {\left(y \cdot z\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(t - a \cdot z\right)\right)\right)}} \]

   4. Applied rewrites 45.4%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 16.0

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

   7. Applied rewrites 16.0%

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

   9. Applied rewrites 77.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - z \cdot y}{t - a \cdot z} \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{t - a \cdot z}\right)\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y, z, -x\right)}{a}}{z}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - a \cdot z} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(a, z, -t\right)}, y, \frac{x}{t - a \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+74}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.26 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(a, -z, t\right)}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(a, z, -t\right)} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.02e+74)
   (/ y a)
   (if (<= z 1.26e+29)
     (/ x (fma a (- z) t))
     (if (<= z 1.2e+109) (* (/ z (fma a z (- t))) y) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.02e+74) {
		tmp = y / a;
	} else if (z <= 1.26e+29) {
		tmp = x / fma(a, -z, t);
	} else if (z <= 1.2e+109) {
		tmp = (z / fma(a, z, -t)) * y;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.02e+74)
		tmp = Float64(y / a);
	elseif (z <= 1.26e+29)
		tmp = Float64(x / fma(a, Float64(-z), t));
	elseif (z <= 1.2e+109)
		tmp = Float64(Float64(z / fma(a, z, Float64(-t))) * y);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.02e+74], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.26e+29], N[(x / N[(a * (-z) + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.2e+109], N[(N[(z / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.02000000000000005e74 or 1.19999999999999994e109 < z

   1. Initial program 54.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 66.5

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

   5. Applied rewrites 66.5%

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

   if -1.02000000000000005e74 < z < 1.26e29

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. lower-*.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 80.6

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

   5. Applied rewrites 80.6%

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

   7. Applied rewrites 80.5%

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

   8. Taylor expanded in x around inf

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

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 75.5

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

   10. Applied rewrites 75.5%

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

   if 1.26e29 < z < 1.19999999999999994e109

   1. Initial program 74.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{t - a \cdot z}} \]
   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 47.9

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

   5. Applied rewrites 47.9%

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

   7. Applied rewrites 63.2%

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

Alternative 6: 66.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+74}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.26 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(a, -z, t\right)}\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+108}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(a, z, -t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.02e+74)
   (/ y a)
   (if (<= z 1.26e+29)
     (/ x (fma a (- z) t))
     (if (<= z 7.6e+108) (* (/ y (fma a z (- t))) z) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.02e+74) {
		tmp = y / a;
	} else if (z <= 1.26e+29) {
		tmp = x / fma(a, -z, t);
	} else if (z <= 7.6e+108) {
		tmp = (y / fma(a, z, -t)) * z;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.02e+74)
		tmp = Float64(y / a);
	elseif (z <= 1.26e+29)
		tmp = Float64(x / fma(a, Float64(-z), t));
	elseif (z <= 7.6e+108)
		tmp = Float64(Float64(y / fma(a, z, Float64(-t))) * z);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.02e+74], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.26e+29], N[(x / N[(a * (-z) + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.6e+108], N[(N[(y / N[(a * z + (-t)), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.02000000000000005e74 or 7.60000000000000015e108 < z

   1. Initial program 54.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 66.5

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

   5. Applied rewrites 66.5%

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

   if -1.02000000000000005e74 < z < 1.26e29

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. lower-*.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 80.6

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

   5. Applied rewrites 80.6%

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

   7. Applied rewrites 80.5%

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

   8. Taylor expanded in x around inf

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

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 75.5

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

   10. Applied rewrites 75.5%

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

   if 1.26e29 < z < 7.60000000000000015e108

   1. Initial program 74.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{t - a \cdot z}} \]
   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 47.9

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

   5. Applied rewrites 47.9%

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

   7. Applied rewrites 58.2%

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

Alternative 7: 90.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+216}:\\ \;\;\;\;\frac{x - z \cdot y}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -5.2e+147)
     t_1
     (if (<= z 2.2e+216) (/ (- x (* z y)) (fma (- z) a 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 <= -5.2e+147) {
		tmp = t_1;
	} else if (z <= 2.2e+216) {
		tmp = (x - (z * y)) / fma(-z, a, 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 <= -5.2e+147)
		tmp = t_1;
	elseif (z <= 2.2e+216)
		tmp = Float64(Float64(x - Float64(z * y)) / fma(Float64(-z), a, t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -5.2e+147], t$95$1, If[LessEqual[z, 2.2e+216], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.1999999999999997e147 or 2.2e216 < z

   1. Initial program 45.0%

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

   3. Taylor expanded in x around 0

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

      1. *-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 66.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 81.4%

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

   if -5.1999999999999997e147 < z < 2.2e216

   1. Initial program 91.4%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 91.4

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

   4. Applied rewrites 91.4%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+147}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+216}:\\ \;\;\;\;\frac{x - z \cdot y}{\mathsf{fma}\left(-z, a, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+216}:\\ \;\;\;\;\frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -5.2e+147)
     t_1
     (if (<= z 2.2e+216) (/ (- x (* z y)) (- t (* a z))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -5.2e+147) {
		tmp = t_1;
	} else if (z <= 2.2e+216) {
		tmp = (x - (z * y)) / (t - (a * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y - (x / z)) / a
    if (z <= (-5.2d+147)) then
        tmp = t_1
    else if (z <= 2.2d+216) then
        tmp = (x - (z * y)) / (t - (a * z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -5.2e+147) {
		tmp = t_1;
	} else if (z <= 2.2e+216) {
		tmp = (x - (z * y)) / (t - (a * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	tmp = 0
	if z <= -5.2e+147:
		tmp = t_1
	elif z <= 2.2e+216:
		tmp = (x - (z * y)) / (t - (a * z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - Float64(x / z)) / a)
	tmp = 0.0
	if (z <= -5.2e+147)
		tmp = t_1;
	elseif (z <= 2.2e+216)
		tmp = Float64(Float64(x - Float64(z * y)) / Float64(t - Float64(a * z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - (x / z)) / a;
	tmp = 0.0;
	if (z <= -5.2e+147)
		tmp = t_1;
	elseif (z <= 2.2e+216)
		tmp = (x - (z * y)) / (t - (a * z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -5.2e+147], t$95$1, If[LessEqual[z, 2.2e+216], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.1999999999999997e147 or 2.2e216 < z

   1. Initial program 45.0%

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

   3. Taylor expanded in x around 0

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

      1. *-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 66.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 81.4%

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

   if -5.1999999999999997e147 < z < 2.2e216

   1. Initial program 91.4%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+147}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+216}:\\ \;\;\;\;\frac{x - z \cdot y}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(a, -z, t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -3.2e+30) t_1 (if (<= z 9.2e+24) (/ x (fma a (- z) 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 <= -3.2e+30) {
		tmp = t_1;
	} else if (z <= 9.2e+24) {
		tmp = x / fma(a, -z, 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 <= -3.2e+30)
		tmp = t_1;
	elseif (z <= 9.2e+24)
		tmp = Float64(x / fma(a, Float64(-z), t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -3.2e+30], t$95$1, If[LessEqual[z, 9.2e+24], N[(x / N[(a * (-z) + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.19999999999999973e30 or 9.1999999999999996e24 < z

   1. Initial program 61.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 76.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 69.5%

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

   if -3.19999999999999973e30 < z < 9.1999999999999996e24

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. lower-*.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 79.4

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

   5. Applied rewrites 79.4%

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

   7. Applied rewrites 79.3%

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

   8. Taylor expanded in x around inf

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

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 75.9

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

   10. Applied rewrites 75.9%

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

Alternative 10: 53.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-184}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{-x}{a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.2e+30)
   (/ y a)
   (if (<= z 7.6e-184) (/ x t) (if (<= z 1.8e+29) (/ (- x) (* a z)) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.2e+30) {
		tmp = y / a;
	} else if (z <= 7.6e-184) {
		tmp = x / t;
	} else if (z <= 1.8e+29) {
		tmp = -x / (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 <= (-3.2d+30)) then
        tmp = y / a
    else if (z <= 7.6d-184) then
        tmp = x / t
    else if (z <= 1.8d+29) then
        tmp = -x / (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 <= -3.2e+30) {
		tmp = y / a;
	} else if (z <= 7.6e-184) {
		tmp = x / t;
	} else if (z <= 1.8e+29) {
		tmp = -x / (a * z);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.2e+30:
		tmp = y / a
	elif z <= 7.6e-184:
		tmp = x / t
	elif z <= 1.8e+29:
		tmp = -x / (a * z)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.2e+30)
		tmp = Float64(y / a);
	elseif (z <= 7.6e-184)
		tmp = Float64(x / t);
	elseif (z <= 1.8e+29)
		tmp = Float64(Float64(-x) / 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 <= -3.2e+30)
		tmp = y / a;
	elseif (z <= 7.6e-184)
		tmp = x / t;
	elseif (z <= 1.8e+29)
		tmp = -x / (a * z);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.2e+30], N[(y / a), $MachinePrecision], If[LessEqual[z, 7.6e-184], N[(x / t), $MachinePrecision], If[LessEqual[z, 1.8e+29], N[((-x) / N[(a * z), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.19999999999999973e30 or 1.79999999999999988e29 < z

   1. Initial program 60.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 58.0

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

   5. Applied rewrites 58.0%

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

   if -3.19999999999999973e30 < z < 7.60000000000000033e-184

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 60.1

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

   5. Applied rewrites 60.1%

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

   if 7.60000000000000033e-184 < z < 1.79999999999999988e29

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   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 95.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 41.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 45.1%

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

Alternative 11: 66.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+74}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(a, -z, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.02e+74) (/ y a) (if (<= z 3e+77) (/ x (fma a (- z) t)) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.02e+74) {
		tmp = y / a;
	} else if (z <= 3e+77) {
		tmp = x / fma(a, -z, t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.02e+74)
		tmp = Float64(y / a);
	elseif (z <= 3e+77)
		tmp = Float64(x / fma(a, Float64(-z), t));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.02e+74], N[(y / a), $MachinePrecision], If[LessEqual[z, 3e+77], N[(x / N[(a * (-z) + t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.02000000000000005e74 or 2.9999999999999998e77 < z

   1. Initial program 56.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 62.7

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

   5. Applied rewrites 62.7%

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

   if -1.02000000000000005e74 < z < 2.9999999999999998e77

   1. Initial program 98.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. *-inverses N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. lower-*.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 80.0

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

   5. Applied rewrites 80.0%

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

   7. Applied rewrites 79.9%

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

   8. Taylor expanded in x around inf

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

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 73.4

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

   10. Applied rewrites 73.4%

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

Alternative 12: 66.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+74}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{t - a \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.02e+74) (/ y a) (if (<= z 3e+77) (/ x (- t (* a z))) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.02e+74) {
		tmp = y / a;
	} else if (z <= 3e+77) {
		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 <= (-1.02d+74)) then
        tmp = y / a
    else if (z <= 3d+77) 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 <= -1.02e+74) {
		tmp = y / a;
	} else if (z <= 3e+77) {
		tmp = x / (t - (a * z));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.02e+74:
		tmp = y / a
	elif z <= 3e+77:
		tmp = x / (t - (a * z))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.02e+74)
		tmp = Float64(y / a);
	elseif (z <= 3e+77)
		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 <= -1.02e+74)
		tmp = y / a;
	elseif (z <= 3e+77)
		tmp = x / (t - (a * z));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.02e+74], N[(y / a), $MachinePrecision], If[LessEqual[z, 3e+77], N[(x / N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.02000000000000005e74 or 2.9999999999999998e77 < z

   1. Initial program 56.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 62.7

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

   5. Applied rewrites 62.7%

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

   if -1.02000000000000005e74 < z < 2.9999999999999998e77

   1. Initial program 98.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 73.4

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

   5. Applied rewrites 73.4%

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

Alternative 13: 55.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.2e+30) (/ y a) (if (<= z 8.5e+25) (/ x t) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.2e+30) {
		tmp = y / a;
	} else if (z <= 8.5e+25) {
		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 <= (-3.2d+30)) then
        tmp = y / a
    else if (z <= 8.5d+25) 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 <= -3.2e+30) {
		tmp = y / a;
	} else if (z <= 8.5e+25) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.2e+30:
		tmp = y / a
	elif z <= 8.5e+25:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.2e+30)
		tmp = Float64(y / a);
	elseif (z <= 8.5e+25)
		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 <= -3.2e+30)
		tmp = y / a;
	elseif (z <= 8.5e+25)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.2e+30], N[(y / a), $MachinePrecision], If[LessEqual[z, 8.5e+25], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.19999999999999973e30 or 8.5000000000000007e25 < z

   1. Initial program 60.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 57.6

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

   5. Applied rewrites 57.6%

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

   if -3.19999999999999973e30 < z < 8.5000000000000007e25

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 50.6

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

   5. Applied rewrites 50.6%

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

Alternative 14: 35.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 35.1

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

5. Applied rewrites 35.1%

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

Developer Target 1: 97.5% 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 2024235 
(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))))