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

Percentage Accurate: 75.1% → 85.0%

Time: 13.4s

Alternatives: 20

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+246}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 -1e+246) (/ z b) (if (<= t_1 2e+292) t_1 (/ 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 tmp;
	if (t_1 <= -1e+246) {
		tmp = z / b;
	} else if (t_1 <= 2e+292) {
		tmp = t_1;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x + ((y * z) / t)) / ((a + 1.0d0) + ((y * b) / t))
    if (t_1 <= (-1d+246)) then
        tmp = z / b
    else if (t_1 <= 2d+292) then
        tmp = t_1
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_1 <= -1e+246) {
		tmp = z / b;
	} else if (t_1 <= 2e+292) {
		tmp = t_1;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t))
	tmp = 0
	if t_1 <= -1e+246:
		tmp = z / b
	elif t_1 <= 2e+292:
		tmp = t_1
	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)))
	tmp = 0.0
	if (t_1 <= -1e+246)
		tmp = Float64(z / b);
	elseif (t_1 <= 2e+292)
		tmp = t_1;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x + ((y * z) / t)) / ((a + 1.0) + ((y * b) / t));
	tmp = 0.0;
	if (t_1 <= -1e+246)
		tmp = z / b;
	elseif (t_1 <= 2e+292)
		tmp = t_1;
	else
		tmp = z / b;
	end
	tmp_2 = 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]}, If[LessEqual[t$95$1, -1e+246], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 2e+292], t$95$1, 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))) < -1.00000000000000007e246 or 2e292 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 22.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 77.6

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

   5. Simplified 77.6%

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

   if -1.00000000000000007e246 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e292

   1. Initial program 91.5%

      \[\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 72.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_3 := \frac{t\_1}{a + 1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+246}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-303}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ (+ a 1.0) (/ (* y b) t))))
        (t_3 (/ t_1 (+ a 1.0))))
   (if (<= t_2 -1e+246)
     (/ z b)
     (if (<= t_2 -4e-303)
       t_3
       (if (<= t_2 0.0)
         (/ x (+ 1.0 (fma y (/ b t) a)))
         (if (<= t_2 2e+292) t_3 (/ z b)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	double t_3 = t_1 / (a + 1.0);
	double tmp;
	if (t_2 <= -1e+246) {
		tmp = z / b;
	} else if (t_2 <= -4e-303) {
		tmp = t_3;
	} else if (t_2 <= 0.0) {
		tmp = x / (1.0 + fma(y, (b / t), a));
	} else if (t_2 <= 2e+292) {
		tmp = t_3;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	t_3 = Float64(t_1 / Float64(a + 1.0))
	tmp = 0.0
	if (t_2 <= -1e+246)
		tmp = Float64(z / b);
	elseif (t_2 <= -4e-303)
		tmp = t_3;
	elseif (t_2 <= 0.0)
		tmp = Float64(x / Float64(1.0 + fma(y, Float64(b / t), a)));
	elseif (t_2 <= 2e+292)
		tmp = t_3;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+246], N[(z / b), $MachinePrecision], If[LessEqual[t$95$2, -4e-303], t$95$3, If[LessEqual[t$95$2, 0.0], N[(x / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+292], t$95$3, N[(z / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.00000000000000007e246 or 2e292 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 22.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 77.6

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

   5. Simplified 77.6%

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

   if -1.00000000000000007e246 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -3.99999999999999972e-303 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e292

   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. lower-+.f64 77.5

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

   5. Simplified 77.5%

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

   if -3.99999999999999972e-303 < (/.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 62.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 70.5

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

   5. Simplified 70.5%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -1 \cdot 10^{+246}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ t_2 := \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+246}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-303}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

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 (/ (fma z (/ y t) x) (+ a 1.0))))
   (if (<= t_1 -1e+246)
     (/ z b)
     (if (<= t_1 -4e-303)
       t_2
       (if (<= t_1 0.0)
         (/ x (+ 1.0 (fma y (/ b t) a)))
         (if (<= t_1 2e+292) 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 = fma(z, (y / t), x) / (a + 1.0);
	double tmp;
	if (t_1 <= -1e+246) {
		tmp = z / b;
	} else if (t_1 <= -4e-303) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = x / (1.0 + fma(y, (b / t), a));
	} else if (t_1 <= 2e+292) {
		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(fma(z, Float64(y / t), x) / Float64(a + 1.0))
	tmp = 0.0
	if (t_1 <= -1e+246)
		tmp = Float64(z / b);
	elseif (t_1 <= -4e-303)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(x / Float64(1.0 + fma(y, Float64(b / t), a)));
	elseif (t_1 <= 2e+292)
		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[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+246], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, -4e-303], t$95$2, If[LessEqual[t$95$1, 0.0], N[(x / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+292], t$95$2, N[(z / b), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.00000000000000007e246 or 2e292 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 22.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 77.6

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

   5. Simplified 77.6%

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

   if -1.00000000000000007e246 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -3.99999999999999972e-303 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e292

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 76.9

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

   5. Simplified 76.9%

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

   if -3.99999999999999972e-303 < (/.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 62.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 70.5

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

   5. Simplified 70.5%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -1 \cdot 10^{+246}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq -4 \cdot 10^{-303}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 0:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}} \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;z \cdot \frac{\frac{y}{t}}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(\frac{b}{t}, y, a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t))) (t_2 (/ t_1 (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_2 (- INFINITY))
     (* z (/ (/ y t) (+ 1.0 (fma y (/ b t) a))))
     (if (<= t_2 2e+292) (/ t_1 (fma (/ b t) y (+ a 1.0))) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / ((a + 1.0) + ((y * b) / t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = z * ((y / t) / (1.0 + fma(y, (b / t), a)));
	} else if (t_2 <= 2e+292) {
		tmp = t_1 / fma((b / t), y, (a + 1.0));
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(a + 1.0) + Float64(Float64(y * b) / t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(z * Float64(Float64(y / t) / Float64(1.0 + fma(y, Float64(b / t), a))));
	elseif (t_2 <= 2e+292)
		tmp = Float64(t_1 / fma(Float64(b / t), y, Float64(a + 1.0)));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 37.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 36.9

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

   4. Applied egg-rr 36.9%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 36.9

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

   7. Simplified 36.9%

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 75.7

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

      11. lift-fma.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 75.7

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

   9. Applied egg-rr 75.7%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e292

   1. Initial program 91.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 89.3

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

   4. Applied egg-rr 89.3%

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

   if 2e292 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 14.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 77.6

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

   5. Simplified 77.6%

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

Alternative 5: 82.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+246}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{z}{t}, y, x\right)}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 -1e+246)
     (/ z b)
     (if (<= t_1 2e+292)
       (/ (fma (/ z t) y x) (fma b (/ y t) (+ a 1.0)))
       (/ 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 tmp;
	if (t_1 <= -1e+246) {
		tmp = z / b;
	} else if (t_1 <= 2e+292) {
		tmp = fma((z / t), y, x) / fma(b, (y / t), (a + 1.0));
	} 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)))
	tmp = 0.0
	if (t_1 <= -1e+246)
		tmp = Float64(z / b);
	elseif (t_1 <= 2e+292)
		tmp = Float64(fma(Float64(z / t), y, x) / fma(b, Float64(y / t), Float64(a + 1.0)));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+246], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 2e+292], N[(N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision] / N[(b * N[(y / t), $MachinePrecision] + N[(a + 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))) < -1.00000000000000007e246 or 2e292 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 22.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 77.6

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

   5. Simplified 77.6%

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

   if -1.00000000000000007e246 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e292

   1. Initial program 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 88.1

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

   4. Applied egg-rr 88.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lift-/.f64 N/A

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

      10. lower-fma.f64 88.5

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

   6. Applied egg-rr 88.5%

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

Alternative 6: 81.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+246}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 -1e+246)
     (/ z b)
     (if (<= t_1 2e+292)
       (/ (fma y (/ z t) x) (fma y (/ b t) (+ a 1.0)))
       (/ 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 tmp;
	if (t_1 <= -1e+246) {
		tmp = z / b;
	} else if (t_1 <= 2e+292) {
		tmp = fma(y, (z / t), x) / fma(y, (b / t), (a + 1.0));
	} 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)))
	tmp = 0.0
	if (t_1 <= -1e+246)
		tmp = Float64(z / b);
	elseif (t_1 <= 2e+292)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(y, Float64(b / t), Float64(a + 1.0)));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(a + 1.0), $MachinePrecision] + N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+246], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 2e+292], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(y * N[(b / t), $MachinePrecision] + N[(a + 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))) < -1.00000000000000007e246 or 2e292 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 22.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 77.6

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

   5. Simplified 77.6%

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

   if -1.00000000000000007e246 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e292

   1. Initial program 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-/.f64 91.5

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/l* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 88.1

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-/l* N/A

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

      21. lower-fma.f64 N/A

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

      22. lower-/.f64 86.7

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

   4. Applied egg-rr 86.7%

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

Alternative 7: 64.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\left(a + 1\right) + \frac{y \cdot b}{t}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+246}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+292}:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (+ a 1.0) (/ (* y b) t)))))
   (if (<= t_1 -1e+246)
     (/ z b)
     (if (<= t_1 2e+292) (/ x (+ 1.0 (fma y (/ b t) a))) (/ 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 tmp;
	if (t_1 <= -1e+246) {
		tmp = z / b;
	} else if (t_1 <= 2e+292) {
		tmp = x / (1.0 + fma(y, (b / t), a));
	} 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)))
	tmp = 0.0
	if (t_1 <= -1e+246)
		tmp = Float64(z / b);
	elseif (t_1 <= 2e+292)
		tmp = Float64(x / Float64(1.0 + fma(y, Float64(b / t), a)));
	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]}, If[LessEqual[t$95$1, -1e+246], N[(z / b), $MachinePrecision], If[LessEqual[t$95$1, 2e+292], N[(x / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $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))) < -1.00000000000000007e246 or 2e292 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 22.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 77.6

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

   5. Simplified 77.6%

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

   if -1.00000000000000007e246 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 2e292

   1. Initial program 91.5%

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

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 67.2

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

   5. Simplified 67.2%

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

Alternative 8: 64.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)\\ \mathbf{if}\;t \leq -1.66 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\frac{a + 1}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}\\ \mathbf{elif}\;t \leq 10^{-152}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(t, \frac{x}{\left(y \cdot z\right) \cdot b}, \frac{1}{b}\right)\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{t\_1}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+43}:\\ \;\;\;\;z \cdot \frac{\frac{y}{t}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ 1.0 (fma y (/ b t) a))))
   (if (<= t -1.66e-25)
     (/ 1.0 (/ (+ a 1.0) (fma y (/ z t) x)))
     (if (<= t 1e-152)
       (* z (fma t (/ x (* (* y z) b)) (/ 1.0 b)))
       (if (<= t 6.4e-50)
         (/ x t_1)
         (if (<= t 2.7e+43)
           (* z (/ (/ y t) t_1))
           (/ (fma z (/ y t) x) (+ a 1.0))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.0 + fma(y, (b / t), a);
	double tmp;
	if (t <= -1.66e-25) {
		tmp = 1.0 / ((a + 1.0) / fma(y, (z / t), x));
	} else if (t <= 1e-152) {
		tmp = z * fma(t, (x / ((y * z) * b)), (1.0 / b));
	} else if (t <= 6.4e-50) {
		tmp = x / t_1;
	} else if (t <= 2.7e+43) {
		tmp = z * ((y / t) / t_1);
	} else {
		tmp = fma(z, (y / t), x) / (a + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.0 + fma(y, Float64(b / t), a))
	tmp = 0.0
	if (t <= -1.66e-25)
		tmp = Float64(1.0 / Float64(Float64(a + 1.0) / fma(y, Float64(z / t), x)));
	elseif (t <= 1e-152)
		tmp = Float64(z * fma(t, Float64(x / Float64(Float64(y * z) * b)), Float64(1.0 / b)));
	elseif (t <= 6.4e-50)
		tmp = Float64(x / t_1);
	elseif (t <= 2.7e+43)
		tmp = Float64(z * Float64(Float64(y / t) / t_1));
	else
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(a + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.66e-25], N[(1.0 / N[(N[(a + 1.0), $MachinePrecision] / N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e-152], N[(z * N[(t * N[(x / N[(N[(y * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.4e-50], N[(x / t$95$1), $MachinePrecision], If[LessEqual[t, 2.7e+43], N[(z * N[(N[(y / t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.6599999999999999e-25

   1. Initial program 85.5%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 87.1

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

   5. Simplified 87.1%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lower-/.f64 88.0

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 88.0

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

   7. Applied egg-rr 88.0%

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

   if -1.6599999999999999e-25 < t < 1.00000000000000007e-152

   1. Initial program 62.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 54.4

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

   4. Applied egg-rr 54.4%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.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. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 41.4

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

   7. Simplified 41.4%

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

   8. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 63.8

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

   10. Simplified 63.8%

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

   if 1.00000000000000007e-152 < t < 6.4e-50

   1. Initial program 88.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 81.9

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

   5. Simplified 81.9%

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

   if 6.4e-50 < t < 2.7000000000000002e43

   1. Initial program 91.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 91.6

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

   4. Applied egg-rr 91.6%

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

   5. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 78.8

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

   7. Simplified 78.8%

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 79.0

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

      11. lift-fma.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 79.0

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

   9. Applied egg-rr 79.0%

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

   if 2.7000000000000002e43 < t

   1. Initial program 89.2%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 82.7

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

   5. Simplified 82.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.66 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\frac{a + 1}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}\\ \mathbf{elif}\;t \leq 10^{-152}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(t, \frac{x}{\left(y \cdot z\right) \cdot b}, \frac{1}{b}\right)\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+43}:\\ \;\;\;\;z \cdot \frac{\frac{y}{t}}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.66 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\frac{a + 1}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-150}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(t, \frac{x}{\left(y \cdot z\right) \cdot b}, \frac{1}{b}\right)\\ \mathbf{elif}\;t \leq 1.96 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(t, \mathsf{fma}\left(b, \frac{y}{t}, a\right), t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -1.66e-25)
   (/ 1.0 (/ (+ a 1.0) (fma y (/ z t) x)))
   (if (<= t 4.6e-150)
     (* z (fma t (/ x (* (* y z) b)) (/ 1.0 b)))
     (if (<= t 1.96e-50)
       (/ x (+ 1.0 (fma y (/ b t) a)))
       (if (<= t 4.6e+35)
         (/ (* y z) (fma t (fma b (/ y t) a) t))
         (/ (fma z (/ y t) x) (+ a 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -1.66e-25) {
		tmp = 1.0 / ((a + 1.0) / fma(y, (z / t), x));
	} else if (t <= 4.6e-150) {
		tmp = z * fma(t, (x / ((y * z) * b)), (1.0 / b));
	} else if (t <= 1.96e-50) {
		tmp = x / (1.0 + fma(y, (b / t), a));
	} else if (t <= 4.6e+35) {
		tmp = (y * z) / fma(t, fma(b, (y / t), a), t);
	} else {
		tmp = fma(z, (y / t), x) / (a + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -1.66e-25)
		tmp = Float64(1.0 / Float64(Float64(a + 1.0) / fma(y, Float64(z / t), x)));
	elseif (t <= 4.6e-150)
		tmp = Float64(z * fma(t, Float64(x / Float64(Float64(y * z) * b)), Float64(1.0 / b)));
	elseif (t <= 1.96e-50)
		tmp = Float64(x / Float64(1.0 + fma(y, Float64(b / t), a)));
	elseif (t <= 4.6e+35)
		tmp = Float64(Float64(y * z) / fma(t, fma(b, Float64(y / t), a), t));
	else
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(a + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -1.66e-25], N[(1.0 / N[(N[(a + 1.0), $MachinePrecision] / N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.6e-150], N[(z * N[(t * N[(x / N[(N[(y * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.96e-50], N[(x / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.6e+35], N[(N[(y * z), $MachinePrecision] / N[(t * N[(b * N[(y / t), $MachinePrecision] + a), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -1.6599999999999999e-25

   1. Initial program 85.5%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 87.1

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

   5. Simplified 87.1%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lower-/.f64 88.0

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 88.0

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

   7. Applied egg-rr 88.0%

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

   if -1.6599999999999999e-25 < t < 4.60000000000000006e-150

   1. Initial program 62.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 54.4

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

   4. Applied egg-rr 54.4%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.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. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 41.4

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

   7. Simplified 41.4%

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

   8. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 63.8

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

   10. Simplified 63.8%

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

   if 4.60000000000000006e-150 < t < 1.9599999999999999e-50

   1. Initial program 88.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 81.9

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

   5. Simplified 81.9%

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

   if 1.9599999999999999e-50 < t < 4.5999999999999996e35

   1. Initial program 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 91.4

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

   4. Applied egg-rr 91.4%

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

   5. Taylor expanded in 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. lower-/.f64 N/A

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

      2. lower-*.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. lower-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. lower-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. lower-/.f64 78.8

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

   7. Simplified 78.8%

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

   if 4.5999999999999996e35 < t

   1. Initial program 89.2%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 82.7

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

   5. Simplified 82.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.66 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\frac{a + 1}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-150}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(t, \frac{x}{\left(y \cdot z\right) \cdot b}, \frac{1}{b}\right)\\ \mathbf{elif}\;t \leq 1.96 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(t, \mathsf{fma}\left(b, \frac{y}{t}, a\right), t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{if}\;t \leq -1.58 \cdot 10^{-86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 10^{-148}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{y \cdot b}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(t, \mathsf{fma}\left(b, \frac{y}{t}, a\right), t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma z (/ y t) x) (+ a 1.0))))
   (if (<= t -1.58e-86)
     t_1
     (if (<= t 1e-148)
       (/ (fma x t (* y z)) (* y b))
       (if (<= t 4.4e-50)
         (/ x (+ 1.0 (fma y (/ b t) a)))
         (if (<= t 4.6e+35) (/ (* y z) (fma t (fma b (/ y t) a) t)) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (y / t), x) / (a + 1.0);
	double tmp;
	if (t <= -1.58e-86) {
		tmp = t_1;
	} else if (t <= 1e-148) {
		tmp = fma(x, t, (y * z)) / (y * b);
	} else if (t <= 4.4e-50) {
		tmp = x / (1.0 + fma(y, (b / t), a));
	} else if (t <= 4.6e+35) {
		tmp = (y * z) / fma(t, fma(b, (y / t), a), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(z, Float64(y / t), x) / Float64(a + 1.0))
	tmp = 0.0
	if (t <= -1.58e-86)
		tmp = t_1;
	elseif (t <= 1e-148)
		tmp = Float64(fma(x, t, Float64(y * z)) / Float64(y * b));
	elseif (t <= 4.4e-50)
		tmp = Float64(x / Float64(1.0 + fma(y, Float64(b / t), a)));
	elseif (t <= 4.6e+35)
		tmp = Float64(Float64(y * z) / fma(t, fma(b, Float64(y / t), a), t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.58e-86], t$95$1, If[LessEqual[t, 1e-148], N[(N[(x * t + N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(y * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.4e-50], N[(x / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.6e+35], N[(N[(y * z), $MachinePrecision] / N[(t * N[(b * N[(y / t), $MachinePrecision] + a), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.58000000000000007e-86 or 4.5999999999999996e35 < t

   1. Initial program 83.3%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 79.6

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

   5. Simplified 79.6%

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

   if -1.58000000000000007e-86 < t < 9.99999999999999936e-149

   1. Initial program 64.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 55.0

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

   4. Applied egg-rr 55.0%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.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. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 45.7

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

   7. Simplified 45.7%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 67.9

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

   10. Simplified 67.9%

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

   if 9.99999999999999936e-149 < t < 4.3999999999999998e-50

   1. Initial program 88.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 81.9

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

   5. Simplified 81.9%

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

   if 4.3999999999999998e-50 < t < 4.5999999999999996e35

   1. Initial program 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 91.4

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

   4. Applied egg-rr 91.4%

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

   5. Taylor expanded in 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. lower-/.f64 N/A

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

      2. lower-*.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. lower-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. lower-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. lower-/.f64 78.8

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

   7. Simplified 78.8%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.58 \cdot 10^{-86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;t \leq 10^{-148}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{y \cdot b}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{y \cdot z}{\mathsf{fma}\left(t, \mathsf{fma}\left(b, \frac{y}{t}, a\right), t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ t_2 := \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a}\\ \mathbf{if}\;a \leq -1.92:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq -3.9 \cdot 10^{-172}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-294}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{y \cdot b}\\ \mathbf{elif}\;a \leq 0.66:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (fma b (/ y t) 1.0))) (t_2 (/ (fma z (/ y t) x) a)))
   (if (<= a -1.92)
     t_2
     (if (<= a -3.9e-172)
       t_1
       (if (<= a 1.3e-294)
         (/ (fma x t (* y z)) (* y b))
         (if (<= a 0.66) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / fma(b, (y / t), 1.0);
	double t_2 = fma(z, (y / t), x) / a;
	double tmp;
	if (a <= -1.92) {
		tmp = t_2;
	} else if (a <= -3.9e-172) {
		tmp = t_1;
	} else if (a <= 1.3e-294) {
		tmp = fma(x, t, (y * z)) / (y * b);
	} else if (a <= 0.66) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / fma(b, Float64(y / t), 1.0))
	t_2 = Float64(fma(z, Float64(y / t), x) / a)
	tmp = 0.0
	if (a <= -1.92)
		tmp = t_2;
	elseif (a <= -3.9e-172)
		tmp = t_1;
	elseif (a <= 1.3e-294)
		tmp = Float64(fma(x, t, Float64(y * z)) / Float64(y * b));
	elseif (a <= 0.66)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(b * N[(y / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[a, -1.92], t$95$2, If[LessEqual[a, -3.9e-172], t$95$1, If[LessEqual[a, 1.3e-294], N[(N[(x * t + N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(y * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.66], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.9199999999999999 or 0.660000000000000031 < a

   1. Initial program 80.0%

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

   3. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 68.9

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

   5. Simplified 68.9%

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

   if -1.9199999999999999 < a < -3.89999999999999973e-172 or 1.3e-294 < a < 0.660000000000000031

   1. Initial program 79.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 76.5

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

   5. Simplified 76.5%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.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. lower-fma.f64 N/A

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

      5. lower-/.f64 61.6

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

   8. Simplified 61.6%

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

   if -3.89999999999999973e-172 < a < 1.3e-294

   1. Initial program 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 71.1

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

   4. Applied egg-rr 71.1%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.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. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 51.4

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

   7. Simplified 51.4%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 68.3

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

   10. Simplified 68.3%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.92:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a}\\ \mathbf{elif}\;a \leq -3.9 \cdot 10^{-172}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-294}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{y \cdot b}\\ \mathbf{elif}\;a \leq 0.66:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ t_2 := \frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a}\\ \mathbf{if}\;a \leq -1.1:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq -5.2 \cdot 10^{-174}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-294}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{y \cdot b}\\ \mathbf{elif}\;a \leq 70:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (fma b (/ y t) 1.0))) (t_2 (/ (fma y (/ z t) x) a)))
   (if (<= a -1.1)
     t_2
     (if (<= a -5.2e-174)
       t_1
       (if (<= a 1.05e-294)
         (/ (fma x t (* y z)) (* y b))
         (if (<= a 70.0) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / fma(b, (y / t), 1.0);
	double t_2 = fma(y, (z / t), x) / a;
	double tmp;
	if (a <= -1.1) {
		tmp = t_2;
	} else if (a <= -5.2e-174) {
		tmp = t_1;
	} else if (a <= 1.05e-294) {
		tmp = fma(x, t, (y * z)) / (y * b);
	} else if (a <= 70.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / fma(b, Float64(y / t), 1.0))
	t_2 = Float64(fma(y, Float64(z / t), x) / a)
	tmp = 0.0
	if (a <= -1.1)
		tmp = t_2;
	elseif (a <= -5.2e-174)
		tmp = t_1;
	elseif (a <= 1.05e-294)
		tmp = Float64(fma(x, t, Float64(y * z)) / Float64(y * b));
	elseif (a <= 70.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(b * N[(y / t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[a, -1.1], t$95$2, If[LessEqual[a, -5.2e-174], t$95$1, If[LessEqual[a, 1.05e-294], N[(N[(x * t + N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(y * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 70.0], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.1000000000000001 or 70 < a

   1. Initial program 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 78.6

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

   4. Applied egg-rr 78.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 66.8

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

   7. Simplified 66.8%

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

   if -1.1000000000000001 < a < -5.2000000000000004e-174 or 1.04999999999999992e-294 < a < 70

   1. Initial program 79.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 76.5

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

   5. Simplified 76.5%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.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. lower-fma.f64 N/A

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

      5. lower-/.f64 61.6

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

   8. Simplified 61.6%

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

   if -5.2000000000000004e-174 < a < 1.04999999999999992e-294

   1. Initial program 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 71.1

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

   4. Applied egg-rr 71.1%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.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. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 51.4

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

   7. Simplified 51.4%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 68.3

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

   10. Simplified 68.3%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a}\\ \mathbf{elif}\;a \leq -5.2 \cdot 10^{-174}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-294}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{y \cdot b}\\ \mathbf{elif}\;a \leq 70:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(b, \frac{y}{t}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, -a, x\right)\\ \mathbf{if}\;a \leq -8.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq -2.95 \cdot 10^{-176}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{-225}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{+55}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma x (- a) x)))
   (if (<= a -8.8e-7)
     (/ x a)
     (if (<= a -2.95e-176)
       t_1
       (if (<= a 1.1e-225)
         (/ z b)
         (if (<= a 1.8e-22) t_1 (if (<= a 1.15e+55) (/ z b) (/ x a))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(x, -a, x);
	double tmp;
	if (a <= -8.8e-7) {
		tmp = x / a;
	} else if (a <= -2.95e-176) {
		tmp = t_1;
	} else if (a <= 1.1e-225) {
		tmp = z / b;
	} else if (a <= 1.8e-22) {
		tmp = t_1;
	} else if (a <= 1.15e+55) {
		tmp = z / b;
	} else {
		tmp = x / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(x, Float64(-a), x)
	tmp = 0.0
	if (a <= -8.8e-7)
		tmp = Float64(x / a);
	elseif (a <= -2.95e-176)
		tmp = t_1;
	elseif (a <= 1.1e-225)
		tmp = Float64(z / b);
	elseif (a <= 1.8e-22)
		tmp = t_1;
	elseif (a <= 1.15e+55)
		tmp = Float64(z / b);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * (-a) + x), $MachinePrecision]}, If[LessEqual[a, -8.8e-7], N[(x / a), $MachinePrecision], If[LessEqual[a, -2.95e-176], t$95$1, If[LessEqual[a, 1.1e-225], N[(z / b), $MachinePrecision], If[LessEqual[a, 1.8e-22], t$95$1, If[LessEqual[a, 1.15e+55], N[(z / b), $MachinePrecision], N[(x / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.8000000000000004e-7 or 1.14999999999999994e55 < a

   1. Initial program 79.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 52.8

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

   5. Simplified 52.8%

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

   6. Taylor expanded in a around inf

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

      1. lower-/.f64 52.5

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

   8. Simplified 52.5%

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

   if -8.8000000000000004e-7 < a < -2.9499999999999998e-176 or 1.1e-225 < a < 1.7999999999999999e-22

   1. Initial program 83.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 51.0

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

   5. Simplified 51.0%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 51.0

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

   8. Simplified 51.0%

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

   if -2.9499999999999998e-176 < a < 1.1e-225 or 1.7999999999999999e-22 < a < 1.14999999999999994e55

   1. Initial program 71.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 53.1

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

   5. Simplified 53.1%

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

Alternative 14: 53.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-128}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+92}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(t, a, t\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ a 1.0))))
   (if (<= t -2.8e-18)
     t_1
     (if (<= t 2.6e-128)
       (/ z b)
       (if (<= t 4e+92) (* y (/ z (fma t a 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 <= -2.8e-18) {
		tmp = t_1;
	} else if (t <= 2.6e-128) {
		tmp = z / b;
	} else if (t <= 4e+92) {
		tmp = y * (z / fma(t, a, 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 <= -2.8e-18)
		tmp = t_1;
	elseif (t <= 2.6e-128)
		tmp = Float64(z / b);
	elseif (t <= 4e+92)
		tmp = Float64(y * Float64(z / fma(t, a, 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, -2.8e-18], t$95$1, If[LessEqual[t, 2.6e-128], N[(z / b), $MachinePrecision], If[LessEqual[t, 4e+92], N[(y * N[(z / N[(t * a + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.80000000000000012e-18 or 4.0000000000000002e92 < t

   1. Initial program 88.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 71.5

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

   5. Simplified 71.5%

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

   if -2.80000000000000012e-18 < t < 2.59999999999999981e-128

   1. Initial program 63.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 60.0

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

   5. Simplified 60.0%

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

   if 2.59999999999999981e-128 < t < 4.0000000000000002e92

   1. Initial program 88.3%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 56.7

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

   5. Simplified 56.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-+.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. lift-/.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lower-/.f64 56.0

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

      16. lift-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 56.0

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

   7. Applied egg-rr 56.0%

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

   8. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 44.8

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

   10. Simplified 44.8%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-128}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+92}:\\ \;\;\;\;y \cdot \frac{z}{\mathsf{fma}\left(t, a, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{-38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ a 1.0))))
   (if (<= t -1.9e-38)
     t_1
     (if (<= t 2.4e+20) (/ (fma x t (* y z)) (* y 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 <= -1.9e-38) {
		tmp = t_1;
	} else if (t <= 2.4e+20) {
		tmp = fma(x, t, (y * z)) / (y * 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 <= -1.9e-38)
		tmp = t_1;
	elseif (t <= 2.4e+20)
		tmp = Float64(fma(x, t, Float64(y * z)) / Float64(y * b));
	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, -1.9e-38], t$95$1, If[LessEqual[t, 2.4e+20], N[(N[(x * t + N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(y * b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.9e-38 or 2.4e20 < t

   1. Initial program 85.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 64.6

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

   5. Simplified 64.6%

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

   if -1.9e-38 < t < 2.4e20

   1. Initial program 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 65.6

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

   4. Applied egg-rr 65.6%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.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. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 43.6

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

   7. Simplified 43.6%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 57.8

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

   10. Simplified 57.8%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, t, y \cdot z\right)}{y \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 41.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a + 1 \leq 0.9999995:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a + 1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, -a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ a 1.0) 0.9999995)
   (/ x a)
   (if (<= (+ a 1.0) 2.0) (fma x (- a) x) (/ x a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a + 1.0) <= 0.9999995) {
		tmp = x / a;
	} else if ((a + 1.0) <= 2.0) {
		tmp = fma(x, -a, x);
	} else {
		tmp = x / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a + 1.0) <= 0.9999995)
		tmp = Float64(x / a);
	elseif (Float64(a + 1.0) <= 2.0)
		tmp = fma(x, Float64(-a), x);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a + 1.0), $MachinePrecision], 0.9999995], N[(x / a), $MachinePrecision], If[LessEqual[N[(a + 1.0), $MachinePrecision], 2.0], N[(x * (-a) + x), $MachinePrecision], N[(x / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 a #s(literal 1 binary64)) < 0.999999500000000041 or 2 < (+.f64 a #s(literal 1 binary64))

   1. Initial program 79.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 52.0

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

   5. Simplified 52.0%

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

   6. Taylor expanded in a around inf

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

      1. lower-/.f64 51.1

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

   8. Simplified 51.1%

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

   if 0.999999500000000041 < (+.f64 a #s(literal 1 binary64)) < 2

   1. Initial program 78.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 39.2

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

   5. Simplified 39.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 39.2

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

   8. Simplified 39.2%

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

Alternative 17: 55.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{a + 1}\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+36}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (+ a 1.0))))
   (if (<= t -3.4e-19) t_1 (if (<= t 1.3e+36) (/ 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 <= -3.4e-19) {
		tmp = t_1;
	} else if (t <= 1.3e+36) {
		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 <= (-3.4d-19)) then
        tmp = t_1
    else if (t <= 1.3d+36) 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 <= -3.4e-19) {
		tmp = t_1;
	} else if (t <= 1.3e+36) {
		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 <= -3.4e-19:
		tmp = t_1
	elif t <= 1.3e+36:
		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 <= -3.4e-19)
		tmp = t_1;
	elseif (t <= 1.3e+36)
		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 <= -3.4e-19)
		tmp = t_1;
	elseif (t <= 1.3e+36)
		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, -3.4e-19], t$95$1, If[LessEqual[t, 1.3e+36], N[(z / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.4000000000000002e-19 or 1.3000000000000001e36 < t

   1. Initial program 87.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 67.5

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

   5. Simplified 67.5%

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

   if -3.4000000000000002e-19 < t < 1.3000000000000001e36

   1. Initial program 70.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 53.5

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

   5. Simplified 53.5%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+36}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 19.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 45.7

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

5. Simplified 45.7%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 20.4

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

8. Simplified 20.4%

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

Alternative 19: 4.1% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.8%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 45.7

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

5. Simplified 45.7%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 20.4

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

8. Simplified 20.4%

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

9. Taylor expanded in a around inf

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 N/A

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

    4. mul-1-neg N/A

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

    5. lower-neg.f64 4.0

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

11. Simplified 4.0%

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

Alternative 20: 19.9% 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 78.8%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 45.7

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

5. Simplified 45.7%

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

6. Taylor expanded in a around 0

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

8. Simplified 21.2%

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

9. Final simplification 21.2%

   \[\leadsto x \]
10. Add Preprocessing

Developer Target 1: 79.9% 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 2024212 
(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))))