Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B

Percentage Accurate: 74.6% → 93.4%

Time: 10.8s

Alternatives: 18

Speedup: 0.2×

Specification

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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), intent (in) :: b
    code = (x + ((y * z) / t)) / ((a + 1.0d0) + ((y * b) / t))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Python

def code(x, y, z, t, a, b):
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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), intent (in) :: b
    code = (x + ((y * z) / t)) / ((a + 1.0d0) + ((y * b) / t))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
}

Python

def code(x, y, z, t, a, b):
	return (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 93.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_2 := \mathsf{fma}\left(\frac{\frac{z}{\mathsf{fma}\left(y, \frac{b}{t}, 1 + a\right)}}{t}, y, \frac{x}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+144}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z}{b} - \frac{\mathsf{fma}\left(-t, \frac{x}{b}, \frac{z}{b \cdot b} \cdot t\right)}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{+285}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a))))
        (t_2
         (fma
          (/ (/ z (fma y (/ b t) (+ 1.0 a))) t)
          y
          (/ x (fma (/ b t) y (+ 1.0 a))))))
   (if (<= t_1 -1e+144)
     t_2
     (if (<= t_1 -5e-311)
       t_1
       (if (<= t_1 0.0)
         (- (/ z b) (/ (fma (- t) (/ x b) (* (/ z (* b b)) t)) y))
         (if (<= t_1 1e+285) t_1 (if (<= t_1 INFINITY) t_2 (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double t_2 = fma(((z / fma(y, (b / t), (1.0 + a))) / t), y, (x / fma((b / t), y, (1.0 + a))));
	double tmp;
	if (t_1 <= -1e+144) {
		tmp = t_2;
	} else if (t_1 <= -5e-311) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (z / b) - (fma(-t, (x / b), ((z / (b * b)) * t)) / y);
	} else if (t_1 <= 1e+285) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_2 = fma(Float64(Float64(z / fma(y, Float64(b / t), Float64(1.0 + a))) / t), y, Float64(x / fma(Float64(b / t), y, Float64(1.0 + a))))
	tmp = 0.0
	if (t_1 <= -1e+144)
		tmp = t_2;
	elseif (t_1 <= -5e-311)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(z / b) - Float64(fma(Float64(-t), Float64(x / b), Float64(Float64(z / Float64(b * b)) * t)) / y));
	elseif (t_1 <= 1e+285)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z / N[(y * N[(b / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * y + N[(x / N[(N[(b / t), $MachinePrecision] * y + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+144], t$95$2, If[LessEqual[t$95$1, -5e-311], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(z / b), $MachinePrecision] - N[(N[((-t) * N[(x / b), $MachinePrecision] + N[(N[(z / N[(b * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+285], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$2, N[(z / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.00000000000000002e144 or 9.9999999999999998e284 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 31.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 87.7%

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

   7. Applied rewrites 97.4%

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

   if -1.00000000000000002e144 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.00000000000023e-311 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999998e284

   1. Initial program 99.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   if -5.00000000000023e-311 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 42.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 29.7

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

   5. Applied rewrites 29.7%

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

   6. Taylor expanded in y around -inf

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{t \cdot x}{b} - -1 \cdot \frac{t \cdot z}{{b}^{2}}}{y} + \color{blue}{\frac{z}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -1 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{\mathsf{fma}\left(y, \frac{b}{t}, 1 + a\right)}}{t}, y, \frac{x}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{z}{b} - \frac{\mathsf{fma}\left(-t, \frac{x}{b}, \frac{z}{b \cdot b} \cdot t\right)}{y}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 10^{+285}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{z}{\mathsf{fma}\left(y, \frac{b}{t}, 1 + a\right)}}{t}, y, \frac{x}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t\_2}, y, \frac{x}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z}{b} - \frac{\mathsf{fma}\left(-t, \frac{x}{b}, \frac{z}{b \cdot b} \cdot t\right)}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{+249}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{y}{t\_2} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a))))
        (t_2 (fma (fma (/ b t) y a) t t)))
   (if (<= t_1 -1e+144)
     (fma (/ z t_2) y (/ x (fma (/ b t) y (+ 1.0 a))))
     (if (<= t_1 -5e-311)
       t_1
       (if (<= t_1 0.0)
         (- (/ z b) (/ (fma (- t) (/ x b) (* (/ z (* b b)) t)) y))
         (if (<= t_1 1e+249)
           t_1
           (if (<= t_1 INFINITY) (* (/ y t_2) z) (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double t_2 = fma(fma((b / t), y, a), t, t);
	double tmp;
	if (t_1 <= -1e+144) {
		tmp = fma((z / t_2), y, (x / fma((b / t), y, (1.0 + a))));
	} else if (t_1 <= -5e-311) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (z / b) - (fma(-t, (x / b), ((z / (b * b)) * t)) / y);
	} else if (t_1 <= 1e+249) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (y / t_2) * z;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_2 = fma(fma(Float64(b / t), y, a), t, t)
	tmp = 0.0
	if (t_1 <= -1e+144)
		tmp = fma(Float64(z / t_2), y, Float64(x / fma(Float64(b / t), y, Float64(1.0 + a))));
	elseif (t_1 <= -5e-311)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(z / b) - Float64(fma(Float64(-t), Float64(x / b), Float64(Float64(z / Float64(b * b)) * t)) / y));
	elseif (t_1 <= 1e+249)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(y / t_2) * z);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+144], N[(N[(z / t$95$2), $MachinePrecision] * y + N[(x / N[(N[(b / t), $MachinePrecision] * y + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -5e-311], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(z / b), $MachinePrecision] - N[(N[((-t) * N[(x / b), $MachinePrecision] + N[(N[(z / N[(b * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+249], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[(y / t$95$2), $MachinePrecision] * z), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.00000000000000002e144

   1. Initial program 48.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 90.8%

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

   if -1.00000000000000002e144 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.00000000000023e-311 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999992e248

   1. Initial program 99.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   if -5.00000000000023e-311 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 42.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 29.7

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

   5. Applied rewrites 29.7%

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

   6. Taylor expanded in y around -inf

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{t \cdot x}{b} - -1 \cdot \frac{t \cdot z}{{b}^{2}}}{y} + \color{blue}{\frac{z}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

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

   if 9.9999999999999992e248 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 15.3%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.3%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -1 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)}, y, \frac{x}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{z}{b} - \frac{\mathsf{fma}\left(-t, \frac{x}{b}, \frac{z}{b \cdot b} \cdot t\right)}{y}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 10^{+249}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_2 := \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z}{b} - \frac{\mathsf{fma}\left(-t, \frac{x}{b}, \frac{z}{b \cdot b} \cdot t\right)}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{+249}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a))))
        (t_2 (* (/ y (fma (fma (/ b t) y a) t t)) z)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -5e-311)
       t_1
       (if (<= t_1 0.0)
         (- (/ z b) (/ (fma (- t) (/ x b) (* (/ z (* b b)) t)) y))
         (if (<= t_1 1e+249) t_1 (if (<= t_1 INFINITY) t_2 (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double t_2 = (y / fma(fma((b / t), y, a), t, t)) * z;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -5e-311) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (z / b) - (fma(-t, (x / b), ((z / (b * b)) * t)) / y);
	} else if (t_1 <= 1e+249) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_2 = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -5e-311)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(z / b) - Float64(fma(Float64(-t), Float64(x / b), Float64(Float64(z / Float64(b * b)) * t)) / y));
	elseif (t_1 <= 1e+249)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -5e-311], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(z / b), $MachinePrecision] - N[(N[((-t) * N[(x / b), $MachinePrecision] + N[(N[(z / N[(b * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+249], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$2, N[(z / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 9.9999999999999992e248 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 14.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.00000000000023e-311 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999992e248

   1. Initial program 99.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   if -5.00000000000023e-311 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 42.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 29.7

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

   5. Applied rewrites 29.7%

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

   6. Taylor expanded in y around -inf

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{t \cdot x}{b} - -1 \cdot \frac{t \cdot z}{{b}^{2}}}{y} + \color{blue}{\frac{z}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{z}{b} - \frac{\mathsf{fma}\left(-t, \frac{x}{b}, \frac{z}{b \cdot b} \cdot t\right)}{y}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 10^{+249}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_2 := \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{b}, \frac{t}{y}, \frac{z}{b}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+249}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a))))
        (t_2 (* (/ y (fma (fma (/ b t) y a) t t)) z)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -5e-311)
       t_1
       (if (<= t_1 0.0)
         (fma (/ x b) (/ t y) (/ z b))
         (if (<= t_1 1e+249) t_1 (if (<= t_1 INFINITY) t_2 (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double t_2 = (y / fma(fma((b / t), y, a), t, t)) * z;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -5e-311) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = fma((x / b), (t / y), (z / b));
	} else if (t_1 <= 1e+249) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_2 = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -5e-311)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(x / b), Float64(t / y), Float64(z / b));
	elseif (t_1 <= 1e+249)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -5e-311], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(x / b), $MachinePrecision] * N[(t / y), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+249], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$2, N[(z / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 9.9999999999999992e248 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 14.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.00000000000023e-311 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999992e248

   1. Initial program 99.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   if -5.00000000000023e-311 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 42.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 21.2

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

   5. Applied rewrites 21.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 70.4%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{b}, \frac{t}{y}, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 10^{+249}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t} + x\\ t_2 := \frac{t\_1}{1 + a}\\ t_3 := \frac{t\_1}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_4 := \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+262}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{b}, \frac{t}{y}, \frac{z}{b}\right)\\ \mathbf{elif}\;t\_3 \leq 10^{+249}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* z y) t) x))
        (t_2 (/ t_1 (+ 1.0 a)))
        (t_3 (/ t_1 (+ (/ (* b y) t) (+ 1.0 a))))
        (t_4 (* (/ y (fma (fma (/ b t) y a) t t)) z)))
   (if (<= t_3 -2e+262)
     t_4
     (if (<= t_3 -5e-311)
       t_2
       (if (<= t_3 0.0)
         (fma (/ x b) (/ t y) (/ z b))
         (if (<= t_3 1e+249) t_2 (if (<= t_3 INFINITY) t_4 (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((z * y) / t) + x;
	double t_2 = t_1 / (1.0 + a);
	double t_3 = t_1 / (((b * y) / t) + (1.0 + a));
	double t_4 = (y / fma(fma((b / t), y, a), t, t)) * z;
	double tmp;
	if (t_3 <= -2e+262) {
		tmp = t_4;
	} else if (t_3 <= -5e-311) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = fma((x / b), (t / y), (z / b));
	} else if (t_3 <= 1e+249) {
		tmp = t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(z * y) / t) + x)
	t_2 = Float64(t_1 / Float64(1.0 + a))
	t_3 = Float64(t_1 / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_4 = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z)
	tmp = 0.0
	if (t_3 <= -2e+262)
		tmp = t_4;
	elseif (t_3 <= -5e-311)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = fma(Float64(x / b), Float64(t / y), Float64(z / b));
	elseif (t_3 <= 1e+249)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+262], t$95$4, If[LessEqual[t$95$3, -5e-311], t$95$2, If[LessEqual[t$95$3, 0.0], N[(N[(x / b), $MachinePrecision] * N[(t / y), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+249], t$95$2, If[LessEqual[t$95$3, Infinity], t$95$4, N[(z / b), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -2e262 or 9.9999999999999992e248 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 19.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.3%

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

   if -2e262 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.00000000000023e-311 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999992e248

   1. Initial program 99.2%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 77.9

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

   5. Applied rewrites 77.9%

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

   if -5.00000000000023e-311 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 42.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 21.2

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

   5. Applied rewrites 21.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 70.4%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -2 \cdot 10^{+262}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{b}, \frac{t}{y}, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 10^{+249}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 41.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+69}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-230}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-289}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, x\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 10^{+249}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a)))))
   (if (<= t_1 -1e+69)
     (/ z b)
     (if (<= t_1 -1e-230)
       (/ x a)
       (if (<= t_1 -4e-289)
         (fma (- x) a x)
         (if (<= t_1 0.0) (/ z b) (if (<= t_1 1e+249) (/ x 1.0) (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double tmp;
	if (t_1 <= -1e+69) {
		tmp = z / b;
	} else if (t_1 <= -1e-230) {
		tmp = x / a;
	} else if (t_1 <= -4e-289) {
		tmp = fma(-x, a, x);
	} else if (t_1 <= 0.0) {
		tmp = z / b;
	} else if (t_1 <= 1e+249) {
		tmp = x / 1.0;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	tmp = 0.0
	if (t_1 <= -1e+69)
		tmp = Float64(z / b);
	elseif (t_1 <= -1e-230)
		tmp = Float64(x / a);
	elseif (t_1 <= -4e-289)
		tmp = fma(Float64(-x), a, x);
	elseif (t_1 <= 0.0)
		tmp = Float64(z / b);
	elseif (t_1 <= 1e+249)
		tmp = Float64(x / 1.0);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+69], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, -1e-230], N[(x / a), $MachinePrecision], If[LessEqual[t$95$1, -4e-289], N[((-x) * a + x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 1e+249], N[(x / 1.0), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.0000000000000001e69 or -4e-289 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0 or 9.9999999999999992e248 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 37.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 54.8

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\frac{z}{b}} \]

   if -1.0000000000000001e69 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.00000000000000005e-230

   1. Initial program 99.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 52.7

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 40.1%

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

   if -1.00000000000000005e-230 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4e-289

   1. Initial program 100.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

      3. lower-+.f64 77.2

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

   5. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\frac{x}{a + 1}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{-1 \cdot \left(a \cdot x\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 75.7%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999992e248

   1. Initial program 99.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

      3. lower-+.f64 60.9

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

   5. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\frac{x}{a + 1}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \frac{x}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.5%

      \[\leadsto \frac{x}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -1 \cdot 10^{+69}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -1 \cdot 10^{-230}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-289}:\\ \;\;\;\;\mathsf{fma}\left(-x, a, x\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 10^{+249}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_2 := \frac{x}{1 + a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-289}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 10^{+249}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a))))
        (t_2 (/ x (+ 1.0 a))))
   (if (<= t_1 -2e+72)
     (fma y (/ z t) x)
     (if (<= t_1 -4e-289)
       t_2
       (if (<= t_1 0.0) (/ z b) (if (<= t_1 1e+249) t_2 (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double t_2 = x / (1.0 + a);
	double tmp;
	if (t_1 <= -2e+72) {
		tmp = fma(y, (z / t), x);
	} else if (t_1 <= -4e-289) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = z / b;
	} else if (t_1 <= 1e+249) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_2 = Float64(x / Float64(1.0 + a))
	tmp = 0.0
	if (t_1 <= -2e+72)
		tmp = fma(y, Float64(z / t), x);
	elseif (t_1 <= -4e-289)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(z / b);
	elseif (t_1 <= 1e+249)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+72], N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, -4e-289], t$95$2, If[LessEqual[t$95$1, 0.0], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 1e+249], t$95$2, N[(z / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.99999999999999989e72

   1. Initial program 65.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 43.7%

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

   if -1.99999999999999989e72 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4e-289 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999992e248

   1. Initial program 99.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

      3. lower-+.f64 58.9

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

   5. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{x}{a + 1}} \]

   if -4e-289 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0 or 9.9999999999999992e248 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 25.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 65.5

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 65.5%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -2 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -4 \cdot 10^{-289}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 10^{+249}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t} + x\\ t_2 := \frac{t\_1}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_3 := \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{+249}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* z y) t) x))
        (t_2 (/ t_1 (+ (/ (* b y) t) (+ 1.0 a))))
        (t_3 (* (/ y (fma (fma (/ b t) y a) t t)) z)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 1e+249)
       (/ t_1 (fma b (/ y t) (+ 1.0 a)))
       (if (<= t_2 INFINITY) t_3 (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((z * y) / t) + x;
	double t_2 = t_1 / (((b * y) / t) + (1.0 + a));
	double t_3 = (y / fma(fma((b / t), y, a), t, t)) * z;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 1e+249) {
		tmp = t_1 / fma(b, (y / t), (1.0 + a));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(z * y) / t) + x)
	t_2 = Float64(t_1 / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_3 = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 1e+249)
		tmp = Float64(t_1 / fma(b, Float64(y / t), Float64(1.0 + a)));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, 1e+249], N[(t$95$1 / N[(b * N[(y / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[(z / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 9.9999999999999992e248 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 14.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999992e248

   1. Initial program 87.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 88.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 88.5

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

   4. Applied rewrites 88.5%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 100.0%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 10^{+249}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\mathsf{fma}\left(b, \frac{y}{t}, 1 + a\right)}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_2 := \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+249}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, 1\right) + a}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a))))
        (t_2 (* (/ y (fma (fma (/ b t) y a) t t)) z)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 1e+249)
       (/ (fma z (/ y t) x) (+ (fma (/ y t) b 1.0) a))
       (if (<= t_1 INFINITY) t_2 (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double t_2 = (y / fma(fma((b / t), y, a), t, t)) * z;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 1e+249) {
		tmp = fma(z, (y / t), x) / (fma((y / t), b, 1.0) + a);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_2 = Float64(Float64(y / fma(fma(Float64(b / t), y, a), t, t)) * z)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 1e+249)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(fma(Float64(y / t), b, 1.0) + a));
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / N[(N[(N[(b / t), $MachinePrecision] * y + a), $MachinePrecision] * t + t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 1e+249], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(y / t), $MachinePrecision] * b + 1.0), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$2, N[(z / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 9.9999999999999992e248 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 14.7%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999992e248

   1. Initial program 87.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 84.9

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

   4. Applied rewrites 84.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 88.3

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

      13. lift-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 88.3

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

   6. Applied rewrites 88.3%

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

   if +inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 0.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 100.0

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 100.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 10^{+249}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\mathsf{fma}\left(\frac{y}{t}, b, 1\right) + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{b}{t}, y, a\right), t, t\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t} + x\\ t_2 := \frac{t\_1}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{b}, \frac{t}{y}, \frac{z}{b}\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+285}:\\ \;\;\;\;\frac{t\_1}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* z y) t) x)) (t_2 (/ t_1 (+ (/ (* b y) t) (+ 1.0 a)))))
   (if (<= t_2 -5e-311)
     (/ (fma z (/ y t) x) (+ 1.0 a))
     (if (<= t_2 0.0)
       (fma (/ x b) (/ t y) (/ z b))
       (if (<= t_2 1e+285) (/ t_1 (+ 1.0 a)) (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((z * y) / t) + x;
	double t_2 = t_1 / (((b * y) / t) + (1.0 + a));
	double tmp;
	if (t_2 <= -5e-311) {
		tmp = fma(z, (y / t), x) / (1.0 + a);
	} else if (t_2 <= 0.0) {
		tmp = fma((x / b), (t / y), (z / b));
	} else if (t_2 <= 1e+285) {
		tmp = t_1 / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(z * y) / t) + x)
	t_2 = Float64(t_1 / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	tmp = 0.0
	if (t_2 <= -5e-311)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(1.0 + a));
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(x / b), Float64(t / y), Float64(z / b));
	elseif (t_2 <= 1e+285)
		tmp = Float64(t_1 / Float64(1.0 + a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-311], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(x / b), $MachinePrecision] * N[(t / y), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+285], N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.00000000000023e-311

   1. Initial program 87.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 68.9

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

   5. Applied rewrites 68.9%

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

   7. Applied rewrites 70.0%

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

   if -5.00000000000023e-311 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 42.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 21.2

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

   5. Applied rewrites 21.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 70.4%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999998e284

   1. Initial program 99.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 79.7

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

   5. Applied rewrites 79.7%

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

   if 9.9999999999999998e284 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 4.5%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 77.0

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{b}, \frac{t}{y}, \frac{z}{b}\right)\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 10^{+285}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t} + x\\ t_2 := \frac{t\_1}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t\_2 \leq 10^{+285}:\\ \;\;\;\;\frac{t\_1}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* z y) t) x)) (t_2 (/ t_1 (+ (/ (* b y) t) (+ 1.0 a)))))
   (if (<= t_2 -5e-311)
     (/ (fma z (/ y t) x) (+ 1.0 a))
     (if (<= t_2 0.0)
       (/ (fma t (/ x y) z) b)
       (if (<= t_2 1e+285) (/ t_1 (+ 1.0 a)) (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((z * y) / t) + x;
	double t_2 = t_1 / (((b * y) / t) + (1.0 + a));
	double tmp;
	if (t_2 <= -5e-311) {
		tmp = fma(z, (y / t), x) / (1.0 + a);
	} else if (t_2 <= 0.0) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t_2 <= 1e+285) {
		tmp = t_1 / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(z * y) / t) + x)
	t_2 = Float64(t_1 / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	tmp = 0.0
	if (t_2 <= -5e-311)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(1.0 + a));
	elseif (t_2 <= 0.0)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t_2 <= 1e+285)
		tmp = Float64(t_1 / Float64(1.0 + a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-311], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$2, 1e+285], N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.00000000000023e-311

   1. Initial program 87.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 68.9

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

   5. Applied rewrites 68.9%

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

   7. Applied rewrites 70.0%

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

   if -5.00000000000023e-311 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 42.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 69.1%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999998e284

   1. Initial program 99.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 79.7

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

   5. Applied rewrites 79.7%

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

   if 9.9999999999999998e284 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 4.5%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 77.0

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 10^{+285}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)}\\ t_2 := \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-311}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t\_1 \leq 10^{+285}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (/ (* b y) t) (+ 1.0 a))))
        (t_2 (/ (fma z (/ y t) x) (+ 1.0 a))))
   (if (<= t_1 -5e-311)
     t_2
     (if (<= t_1 0.0)
       (/ (fma t (/ x y) z) b)
       (if (<= t_1 1e+285) t_2 (/ z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / (((b * y) / t) + (1.0 + a));
	double t_2 = fma(z, (y / t), x) / (1.0 + a);
	double tmp;
	if (t_1 <= -5e-311) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t_1 <= 1e+285) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(Float64(b * y) / t) + Float64(1.0 + a)))
	t_2 = Float64(fma(z, Float64(y / t), x) / Float64(1.0 + a))
	tmp = 0.0
	if (t_1 <= -5e-311)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t_1 <= 1e+285)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision] + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-311], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 1e+285], t$95$2, N[(z / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -5.00000000000023e-311 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999998e284

   1. Initial program 92.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 72.5

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

   5. Applied rewrites 72.5%

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

   7. Applied rewrites 73.8%

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

   if -5.00000000000023e-311 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -0.0

   1. Initial program 42.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 69.1%

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

   if 9.9999999999999998e284 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 4.5%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 77.0

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 77.0%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\frac{b \cdot y}{t} + \left(1 + a\right)} \leq 10^{+285}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -7 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+70}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -7e-48)
     t_1
     (if (<= y 8e+70) (/ x (+ (fma b (/ y t) a) 1.0)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, (x / y), z) / b;
	double tmp;
	if (y <= -7e-48) {
		tmp = t_1;
	} else if (y <= 8e+70) {
		tmp = x / (fma(b, (y / t), a) + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (y <= -7e-48)
		tmp = t_1;
	elseif (y <= 8e+70)
		tmp = Float64(x / Float64(fma(b, Float64(y / t), a) + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -7e-48], t$95$1, If[LessEqual[y, 8e+70], N[(x / N[(N[(b * N[(y / t), $MachinePrecision] + a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.99999999999999982e-48 or 8.00000000000000058e70 < y

   1. Initial program 44.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 62.3%

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

   if -6.99999999999999982e-48 < y < 8.00000000000000058e70

   1. Initial program 93.5%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 70.1

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

   8. Applied rewrites 70.1%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, a\right) + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 60.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{-65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+69}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -4.8e-65) t_1 (if (<= y 8.2e+69) (/ x (+ 1.0 a)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, (x / y), z) / b;
	double tmp;
	if (y <= -4.8e-65) {
		tmp = t_1;
	} else if (y <= 8.2e+69) {
		tmp = x / (1.0 + a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (y <= -4.8e-65)
		tmp = t_1;
	elseif (y <= 8.2e+69)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -4.8e-65], t$95$1, If[LessEqual[y, 8.2e+69], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.8000000000000003e-65 or 8.1999999999999998e69 < y

   1. Initial program 45.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 61.6%

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

   if -4.8000000000000003e-65 < y < 8.1999999999999998e69

   1. Initial program 93.3%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

      3. lower-+.f64 60.5

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

   5. Applied rewrites 60.5%

      \[\leadsto \color{blue}{\frac{x}{a + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+69}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 + a}\\ \mathbf{if}\;t \leq -4 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-114}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ 1.0 a))))
   (if (<= t -4e-5) t_1 (if (<= t 4.6e-114) (/ z b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 + a);
	double tmp;
	if (t <= -4e-5) {
		tmp = t_1;
	} else if (t <= 4.6e-114) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (1.0d0 + a)
    if (t <= (-4d-5)) then
        tmp = t_1
    else if (t <= 4.6d-114) then
        tmp = z / b
    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 b) {
	double t_1 = x / (1.0 + a);
	double tmp;
	if (t <= -4e-5) {
		tmp = t_1;
	} else if (t <= 4.6e-114) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 + a)
	tmp = 0
	if t <= -4e-5:
		tmp = t_1
	elif t <= 4.6e-114:
		tmp = z / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 + a))
	tmp = 0.0
	if (t <= -4e-5)
		tmp = t_1;
	elseif (t <= 4.6e-114)
		tmp = Float64(z / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 + a);
	tmp = 0.0;
	if (t <= -4e-5)
		tmp = t_1;
	elseif (t <= 4.6e-114)
		tmp = z / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4e-5], t$95$1, If[LessEqual[t, 4.6e-114], N[(z / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.00000000000000033e-5 or 4.5999999999999999e-114 < t

   1. Initial program 79.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

      3. lower-+.f64 57.7

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

   5. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\frac{x}{a + 1}} \]

   if -4.00000000000000033e-5 < t < 4.5999999999999999e-114

   1. Initial program 55.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 52.6

         \[\leadsto \color{blue}{\frac{z}{b}} \]

   5. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-5}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-114}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + a}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -34:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 0.75:\\ \;\;\;\;\mathsf{fma}\left(-x, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -34.0) (/ x a) (if (<= a 0.75) (fma (- x) a x) (/ x a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -34.0) {
		tmp = x / a;
	} else if (a <= 0.75) {
		tmp = fma(-x, a, x);
	} else {
		tmp = x / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -34.0)
		tmp = Float64(x / a);
	elseif (a <= 0.75)
		tmp = fma(Float64(-x), a, x);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -34.0], N[(x / a), $MachinePrecision], If[LessEqual[a, 0.75], N[((-x) * a + x), $MachinePrecision], N[(x / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -34 or 0.75 < a

   1. Initial program 69.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 58.2

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 41.4%

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

   if -34 < a < 0.75

   1. Initial program 70.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

      3. lower-+.f64 38.8

         \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

   5. Applied rewrites 38.8%

      \[\leadsto \color{blue}{\frac{x}{a + 1}} \]

   6. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{-1 \cdot \left(a \cdot x\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.5%

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

Alternative 17: 18.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-x, a, x\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (fma (- x) a x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(-x, a, x);
}

Julia

function code(x, y, z, t, a, b)
	return fma(Float64(-x), a, x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[((-x) * a + x), $MachinePrecision]

Derivation

1. Initial program 69.9%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   2. +-commutative N/A

      \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

   3. lower-+.f64 40.7

      \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

5. Applied rewrites 40.7%

   \[\leadsto \color{blue}{\frac{x}{a + 1}} \]

6. Taylor expanded in a around 0

   \[\leadsto x + \color{blue}{-1 \cdot \left(a \cdot x\right)} \]
7. Step-by-step derivation

8. Applied rewrites 20.2%

   \[\leadsto \mathsf{fma}\left(-x, \color{blue}{a}, x\right) \]
9. Add Preprocessing

Alternative 18: 4.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \left(-a\right) \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.9%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
2. Add Preprocessing

3. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{1 + a}} \]

   2. +-commutative N/A

      \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

   3. lower-+.f64 40.7

      \[\leadsto \frac{x}{\color{blue}{a + 1}} \]

5. Applied rewrites 40.7%

   \[\leadsto \color{blue}{\frac{x}{a + 1}} \]

6. Taylor expanded in a around 0

   \[\leadsto x + \color{blue}{-1 \cdot \left(a \cdot x\right)} \]
7. Step-by-step derivation

8. Applied rewrites 20.2%

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

9. Taylor expanded in a around inf

   \[\leadsto -1 \cdot \left(a \cdot \color{blue}{x}\right) \]
10. Step-by-step derivation

11. Applied rewrites 4.2%

    \[\leadsto \left(-a\right) \cdot x \]
12. Add Preprocessing

Developer Target 1: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 \cdot \left(\left(x + \frac{y}{t} \cdot z\right) \cdot \frac{1}{\left(a + 1\right) + \frac{y}{t} \cdot b}\right)\\ \mathbf{if}\;t < -1.3659085366310088 \cdot 10^{-271}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < 3.036967103737246 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (* 1.0 (* (+ x (* (/ y t) z)) (/ 1.0 (+ (+ a 1.0) (* (/ y t) b)))))))
   (if (< t -1.3659085366310088e-271)
     t_1
     (if (< t 3.036967103737246e-130) (/ z b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	double tmp;
	if (t < -1.3659085366310088e-271) {
		tmp = t_1;
	} else if (t < 3.036967103737246e-130) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 * ((x + ((y / t) * z)) * (1.0d0 / ((a + 1.0d0) + ((y / t) * b))))
    if (t < (-1.3659085366310088d-271)) then
        tmp = t_1
    else if (t < 3.036967103737246d-130) then
        tmp = z / b
    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 b) {
	double t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	double tmp;
	if (t < -1.3659085366310088e-271) {
		tmp = t_1;
	} else if (t < 3.036967103737246e-130) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))))
	tmp = 0
	if t < -1.3659085366310088e-271:
		tmp = t_1
	elif t < 3.036967103737246e-130:
		tmp = z / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 * Float64(Float64(x + Float64(Float64(y / t) * z)) * Float64(1.0 / Float64(Float64(a + 1.0) + Float64(Float64(y / t) * b)))))
	tmp = 0.0
	if (t < -1.3659085366310088e-271)
		tmp = t_1;
	elseif (t < 3.036967103737246e-130)
		tmp = Float64(z / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = 1.0 * ((x + ((y / t) * z)) * (1.0 / ((a + 1.0) + ((y / t) * b))));
	tmp = 0.0;
	if (t < -1.3659085366310088e-271)
		tmp = t_1;
	elseif (t < 3.036967103737246e-130)
		tmp = z / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 * N[(N[(x + N[(N[(y / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y / t), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -1.3659085366310088e-271], t$95$1, If[Less[t, 3.036967103737246e-130], N[(z / b), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024254 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1707385670788761/12500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))) (if (< t 1518483551868623/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ z b) (* 1 (* (+ x (* (/ y t) z)) (/ 1 (+ (+ a 1) (* (/ y t) b))))))))

  (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))