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

Percentage Accurate: 75.1% → 92.2%

Time: 14.1s

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: 75.1% 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: 92.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_3 := \frac{1}{\frac{\frac{\mathsf{fma}\left(t, \mathsf{fma}\left(y, \frac{b}{t}, a\right), t\right)}{y}}{z}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-323}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+275}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b))
        (t_2 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_3 (/ 1.0 (/ (/ (fma t (fma y (/ b t) a) t) y) z))))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -5e-323)
       t_2
       (if (<= t_2 0.0)
         t_1
         (if (<= t_2 2e+275) t_2 (if (<= t_2 INFINITY) t_3 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 t_2 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_3 = 1.0 / ((fma(t, fma(y, (b / t), a), t) / y) / z);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -5e-323) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+275) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} 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)
	t_2 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_3 = Float64(1.0 / Float64(Float64(fma(t, fma(y, Float64(b / t), a), t) / y) / z))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -5e-323)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 2e+275)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = t_3;
	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]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(1.0 / N[(N[(N[(t * N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision] + t), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -5e-323], t$95$2, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 2e+275], t$95$2, If[LessEqual[t$95$2, Infinity], t$95$3, t$95$1]]]]]]]]

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 1.99999999999999992e275 < (/.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 24.4%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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 19.9

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

   4. Applied rewrites 19.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 24.4

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

   6. Applied rewrites 51.7%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 38.5

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

   9. Applied rewrites 38.5%

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

   11. Applied rewrites 80.2%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-323 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 1.99999999999999992e275

   1. Initial program 99.1%

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

   if -4.94066e-323 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or +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 24.7%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 36.4%

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

   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 90.5%

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

Alternative 2: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + 1\right) + \frac{y \cdot b}{t}\\ t_2 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ t_3 := \frac{x + \frac{y \cdot z}{t}}{t\_1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(t, \mathsf{fma}\left(b, \frac{y}{t}, a\right), t\right)}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-323}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ a 1.0) (/ (* y b) t)))
        (t_2 (/ (fma t (/ x y) z) b))
        (t_3 (/ (+ x (/ (* y z) t)) t_1)))
   (if (<= t_3 (- INFINITY))
     (* y (/ z (fma t (fma b (/ y t) a) t)))
     (if (<= t_3 -5e-323)
       t_3
       (if (<= t_3 0.0)
         t_2
         (if (<= t_3 5e+292)
           t_3
           (if (<= t_3 INFINITY) (/ (fma (/ z t) y x) t_1) t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + 1.0) + ((y * b) / t);
	double t_2 = fma(t, (x / y), z) / b;
	double t_3 = (x + ((y * z) / t)) / t_1;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = y * (z / fma(t, fma(b, (y / t), a), t));
	} else if (t_3 <= -5e-323) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = t_2;
	} else if (t_3 <= 5e+292) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = fma((z / t), y, x) / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t))
	t_2 = Float64(fma(t, Float64(x / y), z) / b)
	t_3 = Float64(Float64(x + Float64(Float64(y * z) / t)) / t_1)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(y * Float64(z / fma(t, fma(b, Float64(y / t), a), t)));
	elseif (t_3 <= -5e-323)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = t_2;
	elseif (t_3 <= 5e+292)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = Float64(fma(Float64(z / t), y, x) / t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(y * N[(z / N[(t * N[(b * N[(y / t), $MachinePrecision] + a), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-323], t$95$3, If[LessEqual[t$95$3, 0.0], t$95$2, If[LessEqual[t$95$3, 5e+292], t$95$3, If[LessEqual[t$95$3, Infinity], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$2]]]]]]]]

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

   1. Initial program 19.3%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 46.6

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

   4. Applied rewrites 46.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. lower-fma.f64 45.1

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

   6. Applied rewrites 45.1%

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

   7. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 65.1

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

   9. Applied rewrites 65.1%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-323 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.9999999999999996e292

   1. Initial program 99.1%

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

   if -4.94066e-323 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or +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 24.7%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 36.4%

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

   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 90.5%

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

   if 4.9999999999999996e292 < (/.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 24.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 62.3

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

   4. Applied rewrites 62.3%

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

Alternative 3: 88.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + 1\right) + \frac{y \cdot b}{t}\\ t_2 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ t_3 := x + \frac{y \cdot z}{t}\\ t_4 := \frac{t\_3}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ t_5 := \frac{t\_3}{t\_1}\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(t, \mathsf{fma}\left(b, \frac{y}{t}, a\right), t\right)}\\ \mathbf{elif}\;t\_5 \leq -5 \cdot 10^{-323}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ a 1.0) (/ (* y b) t)))
        (t_2 (/ (fma t (/ x y) z) b))
        (t_3 (+ x (/ (* y z) t)))
        (t_4 (/ t_3 (fma b (/ y t) (+ a 1.0))))
        (t_5 (/ t_3 t_1)))
   (if (<= t_5 (- INFINITY))
     (* y (/ z (fma t (fma b (/ y t) a) t)))
     (if (<= t_5 -5e-323)
       t_4
       (if (<= t_5 0.0)
         t_2
         (if (<= t_5 5e+292)
           t_4
           (if (<= t_5 INFINITY) (/ (fma (/ z t) y x) t_1) t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + 1.0) + ((y * b) / t);
	double t_2 = fma(t, (x / y), z) / b;
	double t_3 = x + ((y * z) / t);
	double t_4 = t_3 / fma(b, (y / t), (a + 1.0));
	double t_5 = t_3 / t_1;
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = y * (z / fma(t, fma(b, (y / t), a), t));
	} else if (t_5 <= -5e-323) {
		tmp = t_4;
	} else if (t_5 <= 0.0) {
		tmp = t_2;
	} else if (t_5 <= 5e+292) {
		tmp = t_4;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = fma((z / t), y, x) / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t))
	t_2 = Float64(fma(t, Float64(x / y), z) / b)
	t_3 = Float64(x + Float64(Float64(y * z) / t))
	t_4 = Float64(t_3 / fma(b, Float64(y / t), Float64(a + 1.0)))
	t_5 = Float64(t_3 / t_1)
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(y * Float64(z / fma(t, fma(b, Float64(y / t), a), t)));
	elseif (t_5 <= -5e-323)
		tmp = t_4;
	elseif (t_5 <= 0.0)
		tmp = t_2;
	elseif (t_5 <= 5e+292)
		tmp = t_4;
	elseif (t_5 <= Inf)
		tmp = Float64(fma(Float64(z / t), y, x) / t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$3 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(b * N[(y / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], N[(y * N[(z / N[(t * N[(b * N[(y / t), $MachinePrecision] + a), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, -5e-323], t$95$4, If[LessEqual[t$95$5, 0.0], t$95$2, If[LessEqual[t$95$5, 5e+292], t$95$4, If[LessEqual[t$95$5, Infinity], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$2]]]]]]]]]]

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

   1. Initial program 19.3%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 46.6

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

   4. Applied rewrites 46.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. lower-fma.f64 45.1

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

   6. Applied rewrites 45.1%

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

   7. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 65.1

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

   9. Applied rewrites 65.1%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-323 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.9999999999999996e292

   1. Initial program 99.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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 96.2

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

   4. Applied rewrites 96.2%

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

   if -4.94066e-323 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or +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 24.7%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 36.4%

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

   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 90.5%

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

   if 4.9999999999999996e292 < (/.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 24.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 62.3

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

   4. Applied rewrites 62.3%

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

Alternative 4: 88.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ t_2 := x + \frac{y \cdot z}{t}\\ t_3 := \frac{t\_2}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ t_4 := \frac{t\_2}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(t, \mathsf{fma}\left(b, \frac{y}{t}, a\right), t\right)}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-323}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+292}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b))
        (t_2 (+ x (/ (* y z) t)))
        (t_3 (/ t_2 (fma b (/ y t) (+ a 1.0))))
        (t_4 (/ t_2 (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_4 (- INFINITY))
     (* y (/ z (fma t (fma b (/ y t) a) t)))
     (if (<= t_4 -5e-323)
       t_3
       (if (<= t_4 0.0)
         t_1
         (if (<= t_4 5e+292)
           t_3
           (if (<= t_4 INFINITY)
             (/ (fma y (/ z t) x) (fma y (/ b 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 t_2 = x + ((y * z) / t);
	double t_3 = t_2 / fma(b, (y / t), (a + 1.0));
	double t_4 = t_2 / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = y * (z / fma(t, fma(b, (y / t), a), t));
	} else if (t_4 <= -5e-323) {
		tmp = t_3;
	} else if (t_4 <= 0.0) {
		tmp = t_1;
	} else if (t_4 <= 5e+292) {
		tmp = t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = fma(y, (z / t), x) / fma(y, (b / 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)
	t_2 = Float64(x + Float64(Float64(y * z) / t))
	t_3 = Float64(t_2 / fma(b, Float64(y / t), Float64(a + 1.0)))
	t_4 = Float64(t_2 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(y * Float64(z / fma(t, fma(b, Float64(y / t), a), t)));
	elseif (t_4 <= -5e-323)
		tmp = t_3;
	elseif (t_4 <= 0.0)
		tmp = t_1;
	elseif (t_4 <= 5e+292)
		tmp = t_3;
	elseif (t_4 <= Inf)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(y, Float64(b / t), Float64(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]}, Block[{t$95$2 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(b * N[(y / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(y * N[(z / N[(t * N[(b * N[(y / t), $MachinePrecision] + a), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -5e-323], t$95$3, If[LessEqual[t$95$4, 0.0], t$95$1, If[LessEqual[t$95$4, 5e+292], t$95$3, If[LessEqual[t$95$4, Infinity], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(y * N[(b / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]]

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

   1. Initial program 19.3%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 46.6

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

   4. Applied rewrites 46.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. lower-fma.f64 45.1

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

   6. Applied rewrites 45.1%

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

   7. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 65.1

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

   9. Applied rewrites 65.1%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-323 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.9999999999999996e292

   1. Initial program 99.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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 96.2

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

   4. Applied rewrites 96.2%

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

   if -4.94066e-323 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or +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 24.7%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 36.4%

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

   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 90.5%

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

   if 4.9999999999999996e292 < (/.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 24.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 62.3

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 62.0

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

   4. Applied rewrites 62.0%

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

Alternative 5: 70.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot b}{t}\\ t_2 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ t_3 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + t\_1}\\ t_4 := \mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ t_5 := \frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + t\_1}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{-100}:\\ \;\;\;\;\frac{t\_4}{a + 1}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-323}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{+52}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{t\_4}{\mathsf{fma}\left(y, \frac{b}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* y b) t))
        (t_2 (/ (fma t (/ x y) z) b))
        (t_3 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) t_1)))
        (t_4 (fma z (/ y t) x))
        (t_5 (/ (fma (/ z t) y x) (+ a t_1))))
   (if (<= t_3 -1e-100)
     (/ t_4 (+ a 1.0))
     (if (<= t_3 -5e-323)
       t_5
       (if (<= t_3 0.0)
         t_2
         (if (<= t_3 1e+52)
           t_5
           (if (<= t_3 INFINITY) (/ t_4 (fma y (/ b t) 1.0)) t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * b) / t;
	double t_2 = fma(t, (x / y), z) / b;
	double t_3 = (x + ((y * z) / t)) / ((a + 1.0) + t_1);
	double t_4 = fma(z, (y / t), x);
	double t_5 = fma((z / t), y, x) / (a + t_1);
	double tmp;
	if (t_3 <= -1e-100) {
		tmp = t_4 / (a + 1.0);
	} else if (t_3 <= -5e-323) {
		tmp = t_5;
	} else if (t_3 <= 0.0) {
		tmp = t_2;
	} else if (t_3 <= 1e+52) {
		tmp = t_5;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4 / fma(y, (b / t), 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * b) / t)
	t_2 = Float64(fma(t, Float64(x / y), z) / b)
	t_3 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + t_1))
	t_4 = fma(z, Float64(y / t), x)
	t_5 = Float64(fma(Float64(z / t), y, x) / Float64(a + t_1))
	tmp = 0.0
	if (t_3 <= -1e-100)
		tmp = Float64(t_4 / Float64(a + 1.0));
	elseif (t_3 <= -5e-323)
		tmp = t_5;
	elseif (t_3 <= 0.0)
		tmp = t_2;
	elseif (t_3 <= 1e+52)
		tmp = t_5;
	elseif (t_3 <= Inf)
		tmp = Float64(t_4 / fma(y, Float64(b / t), 1.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(a + t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e-100], N[(t$95$4 / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-323], t$95$5, If[LessEqual[t$95$3, 0.0], t$95$2, If[LessEqual[t$95$3, 1e+52], t$95$5, If[LessEqual[t$95$3, Infinity], N[(t$95$4 / N[(y * N[(b / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]]]

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))) < -1e-100

   1. Initial program 83.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 70.9

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

   5. Applied rewrites 70.9%

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

   if -1e-100 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-323 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999999e51

   1. Initial program 98.3%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.6

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

   4. Applied rewrites 93.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. lower-fma.f64 91.2

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

   6. Applied rewrites 91.2%

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

   7. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 81.2

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

   9. Applied rewrites 81.2%

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

   if -4.94066e-323 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or +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 24.7%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 36.4%

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

   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 90.5%

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

   if 9.9999999999999999e51 < (/.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 75.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 70.7

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

   5. Applied rewrites 70.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -1 \cdot 10^{-100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -5 \cdot 10^{-323}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 10^{+52}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{1}{\frac{\frac{\mathsf{fma}\left(t, \mathsf{fma}\left(y, \frac{b}{t}, a\right), t\right)}{y}}{z}}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-323}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{a + \mathsf{fma}\left(b, \frac{y}{t}, 1\right)}{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b))
        (t_2 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_2 (- INFINITY))
     (/ 1.0 (/ (/ (fma t (fma y (/ b t) a) t) y) z))
     (if (<= t_2 -5e-323)
       t_2
       (if (<= t_2 0.0)
         t_1
         (if (<= t_2 INFINITY)
           (/ 1.0 (/ (+ a (fma b (/ y t) 1.0)) (fma z (/ y t) x)))
           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 t_2 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = 1.0 / ((fma(t, fma(y, (b / t), a), t) / y) / z);
	} else if (t_2 <= -5e-323) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = 1.0 / ((a + fma(b, (y / t), 1.0)) / fma(z, (y / t), x));
	} 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)
	t_2 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(1.0 / Float64(Float64(fma(t, fma(y, Float64(b / t), a), t) / y) / z));
	elseif (t_2 <= -5e-323)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = Float64(1.0 / Float64(Float64(a + fma(b, Float64(y / t), 1.0)) / fma(z, Float64(y / t), x)));
	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]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, (-Infinity)], N[(1.0 / N[(N[(N[(t * N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision] + t), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-323], t$95$2, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, Infinity], N[(1.0 / N[(N[(a + N[(b * N[(y / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

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

   1. Initial program 19.3%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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 17.9

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

   4. Applied rewrites 17.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 19.3

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

   6. Applied rewrites 45.0%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 44.6

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

   9. Applied rewrites 44.6%

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

   11. Applied rewrites 85.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-323

   1. Initial program 99.6%

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

   if -4.94066e-323 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or +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 24.7%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 36.4%

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

   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 90.5%

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

   if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 86.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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 84.4

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

   4. Applied rewrites 84.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 86.5

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

   6. Applied rewrites 91.4%

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

Alternative 7: 83.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-323}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b))
        (t_2 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_2 -5e-323)
     (/ (fma y (/ z t) x) (fma y (/ b t) (+ a 1.0)))
     (if (<= t_2 0.0)
       t_1
       (if (<= t_2 INFINITY)
         (/ (fma (/ z t) y x) (+ a (fma b (/ y t) 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 t_2 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_2 <= -5e-323) {
		tmp = fma(y, (z / t), x) / fma(y, (b / t), (a + 1.0));
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma((z / t), y, x) / (a + fma(b, (y / t), 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)
	t_2 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_2 <= -5e-323)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(y, Float64(b / t), Float64(a + 1.0)));
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = Float64(fma(Float64(z / t), y, x) / Float64(a + fma(b, Float64(y / t), 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]}, Block[{t$95$2 = 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]}, If[LessEqual[t$95$2, -5e-323], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(y * N[(b / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, Infinity], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(a + N[(b * N[(y / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

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))) < -4.94066e-323

   1. Initial program 88.5%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 82.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 81.7

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

   4. Applied rewrites 81.7%

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

   if -4.94066e-323 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or +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 24.7%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 36.4%

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

   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 90.5%

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

   if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < +inf.0

   1. Initial program 86.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 84.7

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

   4. Applied rewrites 84.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. lower-fma.f64 82.5

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

   6. Applied rewrites 82.5%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -5 \cdot 10^{-323}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{a + \mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ t_2 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_3 := \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-323}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b))
        (t_2 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_3 (/ (fma y (/ z t) x) (fma y (/ b t) (+ a 1.0)))))
   (if (<= t_2 -5e-323)
     t_3
     (if (<= t_2 0.0) t_1 (if (<= t_2 INFINITY) t_3 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 t_2 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_3 = fma(y, (z / t), x) / fma(y, (b / t), (a + 1.0));
	double tmp;
	if (t_2 <= -5e-323) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} 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)
	t_2 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_3 = Float64(fma(y, Float64(z / t), x) / fma(y, Float64(b / t), Float64(a + 1.0)))
	tmp = 0.0
	if (t_2 <= -5e-323)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = t_3;
	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]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(y * N[(b / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-323], t$95$3, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, Infinity], t$95$3, t$95$1]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-323 or 0.0 < (/.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 87.6%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 83.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 81.2

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

   4. Applied rewrites 81.2%

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

   if -4.94066e-323 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0 or +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 24.7%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 36.4%

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

   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 90.5%

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

Alternative 9: 68.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{+108}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -2.3e+108)
     t_1
     (if (<= y 9.6e-37) (/ (fma z (/ y t) x) (+ 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 <= -2.3e+108) {
		tmp = t_1;
	} else if (y <= 9.6e-37) {
		tmp = fma(z, (y / t), x) / (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 <= -2.3e+108)
		tmp = t_1;
	elseif (y <= 9.6e-37)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(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, -2.3e+108], t$95$1, If[LessEqual[y, 9.6e-37], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2999999999999999e108 or 9.59999999999999963e-37 < y

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 49.1%

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

   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 71.0%

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

   if -2.2999999999999999e108 < y < 9.59999999999999963e-37

   1. Initial program 90.3%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 75.2

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

   5. Applied rewrites 75.2%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{x}{a + \mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -3.2e+94)
     t_1
     (if (<= y 8.5e-37) (/ x (+ a (fma b (/ y t) 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 <= -3.2e+94) {
		tmp = t_1;
	} else if (y <= 8.5e-37) {
		tmp = x / (a + fma(b, (y / t), 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 <= -3.2e+94)
		tmp = t_1;
	elseif (y <= 8.5e-37)
		tmp = Float64(x / Float64(a + fma(b, Float64(y / t), 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, -3.2e+94], t$95$1, If[LessEqual[y, 8.5e-37], N[(x / N[(a + N[(b * N[(y / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.20000000000000014e94 or 8.5000000000000007e-37 < y

   1. Initial program 48.5%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 50.1%

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

   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 69.8%

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

   if -3.20000000000000014e94 < y < 8.5000000000000007e-37

   1. Initial program 90.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 59.7

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

   5. Applied rewrites 59.7%

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

   7. Applied rewrites 62.4%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{x}{a + \mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -3.2e+94)
     t_1
     (if (<= y 8.5e-37) (/ x (+ 1.0 (fma y (/ b t) 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 <= -3.2e+94) {
		tmp = t_1;
	} else if (y <= 8.5e-37) {
		tmp = x / (1.0 + fma(y, (b / t), 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 <= -3.2e+94)
		tmp = t_1;
	elseif (y <= 8.5e-37)
		tmp = Float64(x / Float64(1.0 + fma(y, Float64(b / t), 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, -3.2e+94], t$95$1, If[LessEqual[y, 8.5e-37], N[(x / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.20000000000000014e94 or 8.5000000000000007e-37 < y

   1. Initial program 48.5%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 50.1%

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

   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 69.8%

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

   if -3.20000000000000014e94 < y < 8.5000000000000007e-37

   1. Initial program 90.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 59.7

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

   5. Applied rewrites 59.7%

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

Alternative 12: 60.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -1.2e-39) t_1 (if (<= y 7.5e-38) (/ x (+ 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 <= -1.2e-39) {
		tmp = t_1;
	} else if (y <= 7.5e-38) {
		tmp = x / (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 <= -1.2e-39)
		tmp = t_1;
	elseif (y <= 7.5e-38)
		tmp = Float64(x / Float64(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, -1.2e-39], t$95$1, If[LessEqual[y, 7.5e-38], N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.20000000000000008e-39 or 7.5e-38 < y

   1. Initial program 54.5%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 53.6%

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

   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 63.4%

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

   if -1.20000000000000008e-39 < y < 7.5e-38

   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 y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 57.5

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

   5. Applied rewrites 57.5%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-146}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -7.8e-45)
   (/ z b)
   (if (<= y 6.8e-146) (/ x a) (if (<= y 2.45e-42) (/ x 1.0) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -7.8e-45) {
		tmp = z / b;
	} else if (y <= 6.8e-146) {
		tmp = x / a;
	} else if (y <= 2.45e-42) {
		tmp = x / 1.0;
	} else {
		tmp = z / b;
	}
	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) :: tmp
    if (y <= (-7.8d-45)) then
        tmp = z / b
    else if (y <= 6.8d-146) then
        tmp = x / a
    else if (y <= 2.45d-42) then
        tmp = x / 1.0d0
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -7.8e-45) {
		tmp = z / b;
	} else if (y <= 6.8e-146) {
		tmp = x / a;
	} else if (y <= 2.45e-42) {
		tmp = x / 1.0;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -7.8e-45:
		tmp = z / b
	elif y <= 6.8e-146:
		tmp = x / a
	elif y <= 2.45e-42:
		tmp = x / 1.0
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -7.8e-45)
		tmp = Float64(z / b);
	elseif (y <= 6.8e-146)
		tmp = Float64(x / a);
	elseif (y <= 2.45e-42)
		tmp = Float64(x / 1.0);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -7.8e-45)
		tmp = z / b;
	elseif (y <= 6.8e-146)
		tmp = x / a;
	elseif (y <= 2.45e-42)
		tmp = x / 1.0;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -7.8e-45], N[(z / b), $MachinePrecision], If[LessEqual[y, 6.8e-146], N[(x / a), $MachinePrecision], If[LessEqual[y, 2.45e-42], N[(x / 1.0), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.7999999999999999e-45 or 2.45e-42 < y

   1. Initial program 54.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 51.1

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

   5. Applied rewrites 51.1%

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

   if -7.7999999999999999e-45 < y < 6.8000000000000001e-146

   1. Initial program 95.4%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 80.8

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 59.8

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

   7. Applied rewrites 59.8%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 41.3%

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

   if 6.8000000000000001e-146 < y < 2.45e-42

   1. Initial program 85.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 83.1

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

   4. Applied rewrites 83.1%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 50.4

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

   7. Applied rewrites 50.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 36.4%

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

Alternative 14: 42.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-140}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{-43}:\\ \;\;\;\;x - x \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -7.8e-45)
   (/ z b)
   (if (<= y 2.3e-140) (/ x a) (if (<= y 1.66e-43) (- x (* x a)) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -7.8e-45) {
		tmp = z / b;
	} else if (y <= 2.3e-140) {
		tmp = x / a;
	} else if (y <= 1.66e-43) {
		tmp = x - (x * a);
	} else {
		tmp = z / b;
	}
	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) :: tmp
    if (y <= (-7.8d-45)) then
        tmp = z / b
    else if (y <= 2.3d-140) then
        tmp = x / a
    else if (y <= 1.66d-43) then
        tmp = x - (x * a)
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -7.8e-45) {
		tmp = z / b;
	} else if (y <= 2.3e-140) {
		tmp = x / a;
	} else if (y <= 1.66e-43) {
		tmp = x - (x * a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -7.8e-45:
		tmp = z / b
	elif y <= 2.3e-140:
		tmp = x / a
	elif y <= 1.66e-43:
		tmp = x - (x * a)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -7.8e-45)
		tmp = Float64(z / b);
	elseif (y <= 2.3e-140)
		tmp = Float64(x / a);
	elseif (y <= 1.66e-43)
		tmp = Float64(x - Float64(x * a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -7.8e-45)
		tmp = z / b;
	elseif (y <= 2.3e-140)
		tmp = x / a;
	elseif (y <= 1.66e-43)
		tmp = x - (x * a);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -7.8e-45], N[(z / b), $MachinePrecision], If[LessEqual[y, 2.3e-140], N[(x / a), $MachinePrecision], If[LessEqual[y, 1.66e-43], N[(x - N[(x * a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.7999999999999999e-45 or 1.66e-43 < y

   1. Initial program 55.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 50.7

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

   5. Applied rewrites 50.7%

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

   if -7.7999999999999999e-45 < y < 2.3000000000000001e-140

   1. Initial program 94.3%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 79.9

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

   4. Applied rewrites 79.9%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 59.2

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

   7. Applied rewrites 59.2%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 40.8%

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

   if 2.3000000000000001e-140 < y < 1.66e-43

   1. Initial program 88.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 85.2

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

   4. Applied rewrites 85.2%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 50.3

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

   7. Applied rewrites 50.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 38.0%

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

Alternative 15: 54.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{+92}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4.3e+92) (/ z b) (if (<= y 1.3e-37) (/ x (+ a 1.0)) (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.3e+92) {
		tmp = z / b;
	} else if (y <= 1.3e-37) {
		tmp = x / (a + 1.0);
	} else {
		tmp = z / b;
	}
	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) :: tmp
    if (y <= (-4.3d+92)) then
        tmp = z / b
    else if (y <= 1.3d-37) then
        tmp = x / (a + 1.0d0)
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.3e+92) {
		tmp = z / b;
	} else if (y <= 1.3e-37) {
		tmp = x / (a + 1.0);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -4.3e+92:
		tmp = z / b
	elif y <= 1.3e-37:
		tmp = x / (a + 1.0)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4.3e+92)
		tmp = Float64(z / b);
	elseif (y <= 1.3e-37)
		tmp = Float64(x / Float64(a + 1.0));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -4.3e+92)
		tmp = z / b;
	elseif (y <= 1.3e-37)
		tmp = x / (a + 1.0);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4.3e+92], N[(z / b), $MachinePrecision], If[LessEqual[y, 1.3e-37], N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.2999999999999998e92 or 1.2999999999999999e-37 < y

   1. Initial program 48.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 56.3

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

   5. Applied rewrites 56.3%

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

   if -4.2999999999999998e92 < y < 1.2999999999999999e-37

   1. Initial program 90.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 52.8

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

   5. Applied rewrites 52.8%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{+92}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-37}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{-43}:\\ \;\;\;\;x - x \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.5e-156) (/ z b) (if (<= y 1.66e-43) (- x (* x a)) (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.5e-156) {
		tmp = z / b;
	} else if (y <= 1.66e-43) {
		tmp = x - (x * a);
	} else {
		tmp = z / b;
	}
	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) :: tmp
    if (y <= (-2.5d-156)) then
        tmp = z / b
    else if (y <= 1.66d-43) then
        tmp = x - (x * a)
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.5e-156) {
		tmp = z / b;
	} else if (y <= 1.66e-43) {
		tmp = x - (x * a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.5e-156:
		tmp = z / b
	elif y <= 1.66e-43:
		tmp = x - (x * a)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.5e-156)
		tmp = Float64(z / b);
	elseif (y <= 1.66e-43)
		tmp = Float64(x - Float64(x * a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.5e-156)
		tmp = z / b;
	elseif (y <= 1.66e-43)
		tmp = x - (x * a);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.5e-156], N[(z / b), $MachinePrecision], If[LessEqual[y, 1.66e-43], N[(x - N[(x * a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.50000000000000004e-156 or 1.66e-43 < y

   1. Initial program 61.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 45.6

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

   5. Applied rewrites 45.6%

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

   if -2.50000000000000004e-156 < y < 1.66e-43

   1. Initial program 90.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 77.8

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

   4. Applied rewrites 77.8%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 57.7

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

   7. Applied rewrites 57.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 26.7%

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

Alternative 17: 18.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ x - x \cdot a \end{array} \]

FPCore

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

C

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

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 - (x * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
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. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 70.1

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

4. Applied rewrites 70.1%

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

5. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 36.0

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

7. Applied rewrites 36.0%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 15.8%

    \[\leadsto x - \color{blue}{x \cdot a} \]
11. Add Preprocessing

Alternative 18: 4.2% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

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 * -a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
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. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 70.1

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

4. Applied rewrites 70.1%

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

5. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 36.0

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

7. Applied rewrites 36.0%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 15.8%

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

11. Taylor expanded in a around inf

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

13. Applied rewrites 4.4%

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

Developer Target 1: 78.3% 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 2024226 
(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))))