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

Percentage Accurate: 74.7% → 92.1%

Time: 13.8s

Alternatives: 15

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.7% 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: 92.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{z}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)} \cdot \frac{y}{t}\\ t_3 := t \cdot \left(\frac{x}{b} - \frac{\mathsf{fma}\left(z, a, z\right)}{b \cdot b}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-319}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\mathsf{fma}\left(\left(-1 - a\right) \cdot t\_3, \frac{t}{y \cdot b}, t\_3\right)}{y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (* (/ z (+ 1.0 (fma y (/ b t) a))) (/ y t)))
        (t_3 (* t (- (/ x b) (/ (fma z a z) (* b b))))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -2e-319)
       t_1
       (if (<= t_1 0.0)
         (+ (/ z b) (/ (fma (* (- -1.0 a) t_3) (/ t (* y b)) t_3) y))
         (if (<= t_1 5e+306) t_1 (if (<= t_1 INFINITY) t_2 (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = (z / (1.0 + fma(y, (b / t), a))) * (y / t);
	double t_3 = t * ((x / b) - (fma(z, a, z) / (b * b)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -2e-319) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (z / b) + (fma(((-1.0 - a) * t_3), (t / (y * b)), t_3) / y);
	} else if (t_1 <= 5e+306) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = Float64(Float64(z / Float64(1.0 + fma(y, Float64(b / t), a))) * Float64(y / t))
	t_3 = Float64(t * Float64(Float64(x / b) - Float64(fma(z, a, z) / Float64(b * b))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -2e-319)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(fma(Float64(Float64(-1.0 - a) * t_3), Float64(t / Float64(y * b)), t_3) / y));
	elseif (t_1 <= 5e+306)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(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$2 = N[(N[(z / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[(N[(x / b), $MachinePrecision] - N[(N[(z * a + z), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -2e-319], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(z / b), $MachinePrecision] + N[(N[(N[(N[(-1.0 - a), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t / N[(y * b), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+306], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$2, N[(z / b), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 4.99999999999999993e306 < (/.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 39.0%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 75.9

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

   4. Applied egg-rr 75.9%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 37.9

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

   7. Simplified 37.9%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r/ N/A

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

      8. associate-+r+ N/A

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

      9. times-frac N/A

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

      10. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 89.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.99998e-319 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.99999999999999993e306

   1. Initial program 98.5%

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

   if -1.99998e-319 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 45.7%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{\left(-1 \cdot \frac{t \cdot x}{b} + -1 \cdot \frac{t \cdot \left(\left(1 + a\right) \cdot \left(-1 \cdot \frac{t \cdot x}{b} - -1 \cdot \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}\right)\right)}{b \cdot y}\right) - -1 \cdot \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2}}}{y} + \frac{z}{b}} \]

   5. Simplified 73.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 95.9

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

   5. Simplified 95.9%

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

Alternative 2: 90.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{z}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)} \cdot \frac{y}{t}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-319}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x}{y \cdot b} - \frac{\mathsf{fma}\left(z, a, z\right)}{b \cdot \left(y \cdot b\right)}, \frac{z}{b}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (* (/ z (+ 1.0 (fma y (/ b t) a))) (/ y t))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -2e-319)
       t_1
       (if (<= t_1 0.0)
         (fma t (- (/ x (* y b)) (/ (fma z a z) (* b (* y b)))) (/ z b))
         (if (<= t_1 5e+306) t_1 (if (<= t_1 INFINITY) t_2 (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = (z / (1.0 + fma(y, (b / t), a))) * (y / t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -2e-319) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = fma(t, ((x / (y * b)) - (fma(z, a, z) / (b * (y * b)))), (z / b));
	} else if (t_1 <= 5e+306) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = Float64(Float64(z / Float64(1.0 + fma(y, Float64(b / t), a))) * Float64(y / t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -2e-319)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = fma(t, Float64(Float64(x / Float64(y * b)) - Float64(fma(z, a, z) / Float64(b * Float64(y * b)))), Float64(z / b));
	elseif (t_1 <= 5e+306)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(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$2 = N[(N[(z / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -2e-319], t$95$1, If[LessEqual[t$95$1, 0.0], N[(t * N[(N[(x / N[(y * b), $MachinePrecision]), $MachinePrecision] - N[(N[(z * a + z), $MachinePrecision] / N[(b * N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+306], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$2, N[(z / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 4.99999999999999993e306 < (/.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 39.0%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 75.9

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

   4. Applied egg-rr 75.9%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 37.9

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

   7. Simplified 37.9%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r/ N/A

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

      8. associate-+r+ N/A

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

      9. times-frac N/A

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

      10. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 89.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.99998e-319 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.99999999999999993e306

   1. Initial program 98.5%

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

   if -1.99998e-319 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 45.7%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(\frac{z}{b} + \frac{t \cdot x}{b \cdot y}\right) - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{z}{b} + \frac{t \cdot x}{b \cdot y}\right) + \left(\mathsf{neg}\left(\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)\right)} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{z}{b} + \left(\frac{t \cdot x}{b \cdot y} + \left(\mathsf{neg}\left(\frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)\right)\right)} \]

      3. sub-neg N/A

         \[\leadsto \frac{z}{b} + \color{blue}{\left(\frac{t \cdot x}{b \cdot y} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{z}{b} + \left(\color{blue}{t \cdot \frac{x}{b \cdot y}} - \frac{t \cdot \left(z \cdot \left(1 + a\right)\right)}{{b}^{2} \cdot y}\right) \]

      5. associate-/l* N/A

         \[\leadsto \frac{z}{b} + \left(t \cdot \frac{x}{b \cdot y} - \color{blue}{t \cdot \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}}\right) \]

      6. distribute-lft-out-- N/A

         \[\leadsto \frac{z}{b} + \color{blue}{t \cdot \left(\frac{x}{b \cdot y} - \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}\right)} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{t \cdot \left(\frac{x}{b \cdot y} - \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}\right) + \frac{z}{b}} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, \frac{x}{b \cdot y} - \frac{z \cdot \left(1 + a\right)}{{b}^{2} \cdot y}, \frac{z}{b}\right)} \]

   5. Simplified 65.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 95.9

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

   5. Simplified 95.9%

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

Alternative 3: 91.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{z}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)} \cdot \frac{y}{t}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-319}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))))
        (t_2 (* (/ z (+ 1.0 (fma y (/ b t) a))) (/ y t))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -2e-319)
       t_1
       (if (<= t_1 0.0)
         (/ (* y z) (fma b y (fma t a t)))
         (if (<= t_1 5e+306) t_1 (if (<= t_1 INFINITY) t_2 (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double t_2 = (z / (1.0 + fma(y, (b / t), a))) * (y / t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -2e-319) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (y * z) / fma(b, y, fma(t, a, t));
	} else if (t_1 <= 5e+306) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_2 = Float64(Float64(z / Float64(1.0 + fma(y, Float64(b / t), a))) * Float64(y / t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -2e-319)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(y * z) / fma(b, y, fma(t, a, t)));
	elseif (t_1 <= 5e+306)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(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$2 = N[(N[(z / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -2e-319], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(y * z), $MachinePrecision] / N[(b * y + N[(t * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+306], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$2, N[(z / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 4.99999999999999993e306 < (/.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 39.0%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 75.9

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

   4. Applied egg-rr 75.9%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 37.9

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

   7. Simplified 37.9%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r/ N/A

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

      8. associate-+r+ N/A

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

      9. times-frac N/A

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

      10. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 89.4%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.99998e-319 or 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.99999999999999993e306

   1. Initial program 98.5%

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

   if -1.99998e-319 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 45.7%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 57.6

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 45.7

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

   7. Simplified 45.7%

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

   8. Taylor expanded in t around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. accelerator-lowering-fma.f64 62.4

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

   10. Simplified 62.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 95.9

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

   5. Simplified 95.9%

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

Alternative 4: 72.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot b}{t}\\ t_2 := x + \frac{y \cdot z}{t}\\ t_3 := \frac{t\_2}{\left(a + 1\right) + t\_1}\\ t_4 := \mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+138}:\\ \;\;\;\;\frac{t\_4}{a + 1}\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-319}:\\ \;\;\;\;\frac{t\_4}{a + t\_1}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{t\_2}{a + 1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{z}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)} \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (* y b) t))
        (t_2 (+ x (/ (* y z) t)))
        (t_3 (/ t_2 (+ (+ a 1.0) t_1)))
        (t_4 (fma y (/ z t) x)))
   (if (<= t_3 -1e+138)
     (/ t_4 (+ a 1.0))
     (if (<= t_3 -2e-319)
       (/ t_4 (+ a t_1))
       (if (<= t_3 0.0)
         (/ (* y z) (fma b y (fma t a t)))
         (if (<= t_3 5e+306)
           (/ t_2 (+ a 1.0))
           (if (<= t_3 INFINITY)
             (* (/ z (+ 1.0 (fma y (/ b t) a))) (/ y t))
             (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * b) / t;
	double t_2 = x + ((y * z) / t);
	double t_3 = t_2 / ((a + 1.0) + t_1);
	double t_4 = fma(y, (z / t), x);
	double tmp;
	if (t_3 <= -1e+138) {
		tmp = t_4 / (a + 1.0);
	} else if (t_3 <= -2e-319) {
		tmp = t_4 / (a + t_1);
	} else if (t_3 <= 0.0) {
		tmp = (y * z) / fma(b, y, fma(t, a, t));
	} else if (t_3 <= 5e+306) {
		tmp = t_2 / (a + 1.0);
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (z / (1.0 + fma(y, (b / t), a))) * (y / t);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * b) / t)
	t_2 = Float64(x + Float64(Float64(y * z) / t))
	t_3 = Float64(t_2 / Float64(Float64(a + 1.0) + t_1))
	t_4 = fma(y, Float64(z / t), x)
	tmp = 0.0
	if (t_3 <= -1e+138)
		tmp = Float64(t_4 / Float64(a + 1.0));
	elseif (t_3 <= -2e-319)
		tmp = Float64(t_4 / Float64(a + t_1));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(y * z) / fma(b, y, fma(t, a, t)));
	elseif (t_3 <= 5e+306)
		tmp = Float64(t_2 / Float64(a + 1.0));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(z / Float64(1.0 + fma(y, Float64(b / t), a))) * Float64(y / t));
	else
		tmp = Float64(z / b);
	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[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(N[(a + 1.0), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+138], N[(t$95$4 / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -2e-319], N[(t$95$4 / N[(a + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(y * z), $MachinePrecision] / N[(b * y + N[(t * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+306], N[(t$95$2 / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(z / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1e138

   1. Initial program 47.7%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 73.7

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

   4. Applied egg-rr 73.7%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 69.2%

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

   if -1e138 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.99998e-319

   1. Initial program 97.0%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 92.0

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

   4. Applied egg-rr 92.0%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 81.4

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

   7. Simplified 81.4%

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

   if -1.99998e-319 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 0.0

   1. Initial program 45.7%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 57.6

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 45.7

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

   7. Simplified 45.7%

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

   8. Taylor expanded in t around 0

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. accelerator-lowering-fma.f64 62.4

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

   10. Simplified 62.4%

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

   if 0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 4.99999999999999993e306

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 74.3

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

   5. Simplified 74.3%

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

   if 4.99999999999999993e306 < (/.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 57.0%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 92.5

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

   4. Applied egg-rr 92.5%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 41.3

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

   7. Simplified 41.3%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r/ N/A

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

      8. associate-+r+ N/A

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

      9. times-frac N/A

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

      10. *-lowering-*.f64 N/A

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

   9. Applied egg-rr 99.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 95.9

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

   5. Simplified 95.9%

      \[\leadsto \color{blue}{\frac{z}{b}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -1 \cdot 10^{+138}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -2 \cdot 10^{-319}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a + \frac{y \cdot b}{t}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(t, a, t\right)\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\frac{z}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)} \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a + \mathsf{fma}\left(y, \frac{b}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t))) INFINITY)
   (/ (fma y (/ z t) x) (+ a (fma y (/ b t) 1.0)))
   (/ z b)))

C

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

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t))) <= Inf)
		tmp = Float64(fma(y, Float64(z / t), x) / Float64(a + fma(y, Float64(b / t), 1.0)));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

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

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))) < +inf.0

   1. Initial program 80.4%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 80.5

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

   4. Applied egg-rr 80.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 95.9

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

   5. Simplified 95.9%

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

Alternative 6: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a + 1}\\ \mathbf{if}\;t \leq -1.65 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2.2 \cdot 10^{-260}:\\ \;\;\;\;t \cdot \left(\frac{x}{y \cdot b} + \frac{z}{t \cdot b}\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-111}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(y, b, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma y (/ z t) x) (+ a 1.0))))
   (if (<= t -1.65e-16)
     t_1
     (if (<= t -2.2e-260)
       (* t (+ (/ x (* y b)) (/ z (* t b))))
       (if (<= t 8e-111) (* y (/ z (fma a t (fma y b t)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (z / t), x) / (a + 1.0);
	double tmp;
	if (t <= -1.65e-16) {
		tmp = t_1;
	} else if (t <= -2.2e-260) {
		tmp = t * ((x / (y * b)) + (z / (t * b)));
	} else if (t <= 8e-111) {
		tmp = y * (z / fma(a, t, fma(y, b, t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(y, Float64(z / t), x) / Float64(a + 1.0))
	tmp = 0.0
	if (t <= -1.65e-16)
		tmp = t_1;
	elseif (t <= -2.2e-260)
		tmp = Float64(t * Float64(Float64(x / Float64(y * b)) + Float64(z / Float64(t * b))));
	elseif (t <= 8e-111)
		tmp = Float64(y * Float64(z / fma(a, t, fma(y, b, t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.65e-16], t$95$1, If[LessEqual[t, -2.2e-260], N[(t * N[(N[(x / N[(y * b), $MachinePrecision]), $MachinePrecision] + N[(z / N[(t * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-111], N[(y * N[(z / N[(a * t + N[(y * b + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.64999999999999994e-16 or 8.00000000000000071e-111 < t

   1. Initial program 77.3%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 89.2

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

   4. Applied egg-rr 89.2%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 74.1%

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

   if -1.64999999999999994e-16 < t < -2.20000000000000017e-260

   1. Initial program 67.0%

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

   3. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 48.3

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

   5. Simplified 48.3%

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

   6. Taylor expanded in t around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-lowering-+.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 71.9

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

   8. Simplified 71.9%

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

   if -2.20000000000000017e-260 < t < 8.00000000000000071e-111

   1. Initial program 67.1%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 54.7

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

   4. Applied egg-rr 54.7%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 46.2

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

   7. Simplified 46.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 48.0

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

   9. Applied egg-rr 48.0%

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

   10. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-lft-identity N/A

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

       4. +-commutative N/A

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

       5. associate-+l+ N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. accelerator-lowering-fma.f64 70.7

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

   12. Simplified 70.7%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;t \leq -2.2 \cdot 10^{-260}:\\ \;\;\;\;t \cdot \left(\frac{x}{y \cdot b} + \frac{z}{t \cdot b}\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-111}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(y, b, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a + 1}\\ \mathbf{if}\;t \leq -1.04 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-208}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{y \cdot b}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-111}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(y, b, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma y (/ z t) x) (+ a 1.0))))
   (if (<= t -1.04e-16)
     t_1
     (if (<= t 1.7e-208)
       (/ (fma x t (* y z)) (* y b))
       (if (<= t 2.2e-111) (* y (/ z (fma a t (fma y b t)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (z / t), x) / (a + 1.0);
	double tmp;
	if (t <= -1.04e-16) {
		tmp = t_1;
	} else if (t <= 1.7e-208) {
		tmp = fma(x, t, (y * z)) / (y * b);
	} else if (t <= 2.2e-111) {
		tmp = y * (z / fma(a, t, fma(y, b, t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(y, Float64(z / t), x) / Float64(a + 1.0))
	tmp = 0.0
	if (t <= -1.04e-16)
		tmp = t_1;
	elseif (t <= 1.7e-208)
		tmp = Float64(fma(x, t, Float64(y * z)) / Float64(y * b));
	elseif (t <= 2.2e-111)
		tmp = Float64(y * Float64(z / fma(a, t, fma(y, b, t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.04e-16], t$95$1, If[LessEqual[t, 1.7e-208], N[(N[(x * t + N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(y * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.2e-111], N[(y * N[(z / N[(a * t + N[(y * b + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.04000000000000001e-16 or 2.2e-111 < t

   1. Initial program 77.3%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 89.2

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

   4. Applied egg-rr 89.2%

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

   5. Taylor expanded in y around 0

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

   7. Simplified 74.1%

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

   if -1.04000000000000001e-16 < t < 1.7e-208

   1. Initial program 62.0%

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

   3. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 43.1

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

   5. Simplified 43.1%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 68.0

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

   8. Simplified 68.0%

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

   if 1.7e-208 < t < 2.2e-111

   1. Initial program 79.1%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 61.6

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

   4. Applied egg-rr 61.6%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 61.8

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

   7. Simplified 61.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 59.4

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

   9. Applied egg-rr 59.4%

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

   10. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-lft-identity N/A

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

       4. +-commutative N/A

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

       5. associate-+l+ N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. accelerator-lowering-fma.f64 72.6

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

   12. Simplified 72.6%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.04 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-208}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{y \cdot b}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-111}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(y, b, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+102}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(y, b, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -2.9e+28)
   (/ x (+ 1.0 (fma y (/ b t) a)))
   (if (<= x 3.5e+102)
     (* y (/ z (fma a t (fma y b t))))
     (/ x (fma b (/ y t) (+ a 1.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -2.9e+28) {
		tmp = x / (1.0 + fma(y, (b / t), a));
	} else if (x <= 3.5e+102) {
		tmp = y * (z / fma(a, t, fma(y, b, t)));
	} else {
		tmp = x / fma(b, (y / t), (a + 1.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -2.9e+28)
		tmp = Float64(x / Float64(1.0 + fma(y, Float64(b / t), a)));
	elseif (x <= 3.5e+102)
		tmp = Float64(y * Float64(z / fma(a, t, fma(y, b, t))));
	else
		tmp = Float64(x / fma(b, Float64(y / t), Float64(a + 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -2.9e+28], N[(x / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.5e+102], N[(y * N[(z / N[(a * t + N[(y * b + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(b * N[(y / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9000000000000001e28

   1. Initial program 77.5%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.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. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 71.0

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

   5. Simplified 71.0%

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

   if -2.9000000000000001e28 < x < 3.50000000000000011e102

   1. Initial program 70.8%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 71.7

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

   4. Applied egg-rr 71.7%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 47.0

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

   7. Simplified 47.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 51.7

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

   9. Applied egg-rr 51.7%

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

   10. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-lft-identity N/A

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

       4. +-commutative N/A

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

       5. associate-+l+ N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. accelerator-lowering-fma.f64 63.4

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

   12. Simplified 63.4%

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

   if 3.50000000000000011e102 < x

   1. Initial program 73.1%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.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. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 77.9

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

   5. Simplified 77.9%

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 77.9

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

   7. Applied egg-rr 77.9%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+28}:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+102}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(y, b, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+101}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(y, b, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ 1.0 (fma y (/ b t) a)))))
   (if (<= x -3.2e+29)
     t_1
     (if (<= x 9e+101) (* y (/ z (fma a t (fma y b t)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 + fma(y, (b / t), a));
	double tmp;
	if (x <= -3.2e+29) {
		tmp = t_1;
	} else if (x <= 9e+101) {
		tmp = y * (z / fma(a, t, fma(y, b, t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 + fma(y, Float64(b / t), a)))
	tmp = 0.0
	if (x <= -3.2e+29)
		tmp = t_1;
	elseif (x <= 9e+101)
		tmp = Float64(y * Float64(z / fma(a, t, fma(y, b, t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e+29], t$95$1, If[LessEqual[x, 9e+101], N[(y * N[(z / N[(a * t + N[(y * b + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.19999999999999987e29 or 9.0000000000000004e101 < x

   1. Initial program 75.9%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.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. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 73.6

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

   5. Simplified 73.6%

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

   if -3.19999999999999987e29 < x < 9.0000000000000004e101

   1. Initial program 70.8%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 71.7

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

   4. Applied egg-rr 71.7%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 47.0

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

   7. Simplified 47.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 51.7

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

   9. Applied egg-rr 51.7%

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

   10. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-lft-identity N/A

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

       4. +-commutative N/A

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

       5. associate-+l+ N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. accelerator-lowering-fma.f64 63.4

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

   12. Simplified 63.4%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+101}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(y, b, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -4.15 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-208}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{y \cdot b}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-109}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(y, b, t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ a 1.0))))
   (if (<= t -4.15e-16)
     t_1
     (if (<= t 8e-208)
       (/ (fma x t (* y z)) (* y b))
       (if (<= t 1.65e-109) (* y (/ z (fma y b t))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a + 1.0);
	double tmp;
	if (t <= -4.15e-16) {
		tmp = t_1;
	} else if (t <= 8e-208) {
		tmp = fma(x, t, (y * z)) / (y * b);
	} else if (t <= 1.65e-109) {
		tmp = y * (z / fma(y, b, t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a + 1.0))
	tmp = 0.0
	if (t <= -4.15e-16)
		tmp = t_1;
	elseif (t <= 8e-208)
		tmp = Float64(fma(x, t, Float64(y * z)) / Float64(y * b));
	elseif (t <= 1.65e-109)
		tmp = Float64(y * Float64(z / fma(y, b, t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.15e-16], t$95$1, If[LessEqual[t, 8e-208], N[(N[(x * t + N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(y * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.65e-109], N[(y * N[(z / N[(y * b + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.14999999999999989e-16 or 1.64999999999999995e-109 < t

   1. Initial program 77.3%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 57.5

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

   5. Simplified 57.5%

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

   if -4.14999999999999989e-16 < t < 8.0000000000000008e-208

   1. Initial program 62.0%

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

   3. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 43.1

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

   5. Simplified 43.1%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 68.0

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

   8. Simplified 68.0%

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

   if 8.0000000000000008e-208 < t < 1.64999999999999995e-109

   1. Initial program 79.1%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 61.6

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

   4. Applied egg-rr 61.6%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 61.8

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

   7. Simplified 61.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 59.4

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

   9. Applied egg-rr 59.4%

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

   10. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. accelerator-lowering-fma.f64 54.5

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

   12. Simplified 54.5%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.15 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-208}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{y \cdot b}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-109}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(y, b, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-89}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(y, b, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ a 1.0))))
   (if (<= t -3.5e+34)
     t_1
     (if (<= t 2.3e-89) (* y (/ z (fma a t (fma y b t)))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a + 1.0);
	double tmp;
	if (t <= -3.5e+34) {
		tmp = t_1;
	} else if (t <= 2.3e-89) {
		tmp = y * (z / fma(a, t, fma(y, b, t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a + 1.0))
	tmp = 0.0
	if (t <= -3.5e+34)
		tmp = t_1;
	elseif (t <= 2.3e-89)
		tmp = Float64(y * Float64(z / fma(a, t, fma(y, b, t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.5e+34], t$95$1, If[LessEqual[t, 2.3e-89], N[(y * N[(z / N[(a * t + N[(y * b + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.49999999999999998e34 or 2.3e-89 < t

   1. Initial program 76.5%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 60.2

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

   5. Simplified 60.2%

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

   if -3.49999999999999998e34 < t < 2.3e-89

   1. Initial program 69.0%

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-+l+ N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 57.6

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 44.0

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

   7. Simplified 44.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. associate-*r/ N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. /-lowering-/.f64 45.8

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

   9. Applied egg-rr 45.8%

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

   10. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. *-lft-identity N/A

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

       4. +-commutative N/A

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

       5. associate-+l+ N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. +-commutative N/A

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

       8. *-commutative N/A

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

       9. accelerator-lowering-fma.f64 63.0

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

   12. Simplified 63.0%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-89}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(a, t, \mathsf{fma}\left(y, b, t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a + 1 \leq -4 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a + 1 \leq 2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ a 1.0) -4e+19) (/ x a) (if (<= (+ a 1.0) 2.0) x (/ x a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a + 1.0) <= -4e+19) {
		tmp = x / a;
	} else if ((a + 1.0) <= 2.0) {
		tmp = x;
	} else {
		tmp = x / a;
	}
	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 ((a + 1.0d0) <= (-4d+19)) then
        tmp = x / a
    else if ((a + 1.0d0) <= 2.0d0) then
        tmp = x
    else
        tmp = x / a
    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 ((a + 1.0) <= -4e+19) {
		tmp = x / a;
	} else if ((a + 1.0) <= 2.0) {
		tmp = x;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a + 1.0) <= -4e+19:
		tmp = x / a
	elif (a + 1.0) <= 2.0:
		tmp = x
	else:
		tmp = x / a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a + 1.0) <= -4e+19)
		tmp = Float64(x / a);
	elseif (Float64(a + 1.0) <= 2.0)
		tmp = x;
	else
		tmp = Float64(x / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((a + 1.0) <= -4e+19)
		tmp = x / a;
	elseif ((a + 1.0) <= 2.0)
		tmp = x;
	else
		tmp = x / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a + 1.0), $MachinePrecision], -4e+19], N[(x / a), $MachinePrecision], If[LessEqual[N[(a + 1.0), $MachinePrecision], 2.0], x, N[(x / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 a #s(literal 1 binary64)) < -4e19 or 2 < (+.f64 a #s(literal 1 binary64))

   1. Initial program 73.0%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.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. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 56.7

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

   5. Simplified 56.7%

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

   6. Taylor expanded in a around inf

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

      1. /-lowering-/.f64 49.0

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

   8. Simplified 49.0%

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

   if -4e19 < (+.f64 a #s(literal 1 binary64)) < 2

   1. Initial program 72.8%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   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. /-lowering-/.f64 N/A

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

      2. +-lowering-+.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. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 44.3

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

   5. Simplified 44.3%

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

   6. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 42.5

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

   8. Simplified 42.5%

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

   9. Taylor expanded in b around 0

      \[\leadsto \color{blue}{x} \]
   10. Step-by-step derivation

   11. Simplified 30.5%

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

Alternative 13: 55.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ a 1.0))))
   (if (<= t -2.8e+17) t_1 (if (<= t 5.4e-52) (/ z b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a + 1.0);
	double tmp;
	if (t <= -2.8e+17) {
		tmp = t_1;
	} else if (t <= 5.4e-52) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (a + 1.0d0)
    if (t <= (-2.8d+17)) then
        tmp = t_1
    else if (t <= 5.4d-52) then
        tmp = z / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (a + 1.0);
	double tmp;
	if (t <= -2.8e+17) {
		tmp = t_1;
	} else if (t <= 5.4e-52) {
		tmp = z / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (a + 1.0)
	tmp = 0
	if t <= -2.8e+17:
		tmp = t_1
	elif t <= 5.4e-52:
		tmp = z / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(a + 1.0))
	tmp = 0.0
	if (t <= -2.8e+17)
		tmp = t_1;
	elseif (t <= 5.4e-52)
		tmp = Float64(z / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (a + 1.0);
	tmp = 0.0;
	if (t <= -2.8e+17)
		tmp = t_1;
	elseif (t <= 5.4e-52)
		tmp = z / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.8e+17], t$95$1, If[LessEqual[t, 5.4e-52], N[(z / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.8e17 or 5.40000000000000019e-52 < t

   1. Initial program 78.5%

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

   3. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.f64 61.5

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

   5. Simplified 61.5%

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

   if -2.8e17 < t < 5.40000000000000019e-52

   1. Initial program 67.5%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 54.4

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

   5. Simplified 54.4%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.45e+15) (/ x a) (if (<= a 1.5e+95) (/ z b) (/ x a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.45e+15) {
		tmp = x / a;
	} else if (a <= 1.5e+95) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	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 (a <= (-1.45d+15)) then
        tmp = x / a
    else if (a <= 1.5d+95) then
        tmp = z / b
    else
        tmp = x / a
    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 (a <= -1.45e+15) {
		tmp = x / a;
	} else if (a <= 1.5e+95) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -1.45e+15:
		tmp = x / a
	elif a <= 1.5e+95:
		tmp = z / b
	else:
		tmp = x / a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.45e+15)
		tmp = Float64(x / a);
	elseif (a <= 1.5e+95)
		tmp = Float64(z / b);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -1.45e+15)
		tmp = x / a;
	elseif (a <= 1.5e+95)
		tmp = z / b;
	else
		tmp = x / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.45e+15], N[(x / a), $MachinePrecision], If[LessEqual[a, 1.5e+95], N[(z / b), $MachinePrecision], N[(x / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.45e15 or 1.49999999999999996e95 < a

   1. Initial program 74.5%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

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

      2. +-lowering-+.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. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 61.1

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

   5. Simplified 61.1%

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

   6. Taylor expanded in a around inf

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

      1. /-lowering-/.f64 55.0

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

   8. Simplified 55.0%

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

   if -1.45e15 < a < 1.49999999999999996e95

   1. Initial program 71.6%

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

   3. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 46.4

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

   5. Simplified 46.4%

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

Alternative 15: 19.8% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 72.9%

   \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
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. /-lowering-/.f64 N/A

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

   2. +-lowering-+.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. accelerator-lowering-fma.f64 N/A

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

   7. /-lowering-/.f64 50.7

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

5. Simplified 50.7%

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

6. Taylor expanded in a around 0

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

   1. /-lowering-/.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-/l* N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. /-lowering-/.f64 27.5

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

8. Simplified 27.5%

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

9. Taylor expanded in b around 0

   \[\leadsto \color{blue}{x} \]
10. Step-by-step derivation

11. Simplified 16.7%

    \[\leadsto \color{blue}{x} \]
12. Add Preprocessing

Developer Target 1: 78.6% 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 2024196 
(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))))