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

Percentage Accurate: 74.9% → 93.6%

Time: 9.9s

Alternatives: 14

Speedup: 0.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 + a\right) + \frac{b \cdot y}{t}\\ t_2 := \frac{\frac{z \cdot y}{t} + x}{t\_1}\\ t_3 := \frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\frac{x}{b} \cdot t}{y}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{1}{\frac{t}{z \cdot y}} + x}{t\_1}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ 1.0 a) (/ (* b y) t)))
        (t_2 (/ (+ (/ (* z y) t) x) t_1))
        (t_3 (* (/ y (fma b y (fma a t t))) z)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -5e-324)
       t_2
       (if (<= t_2 0.0)
         (+ (/ z b) (/ (* (/ x b) t) y))
         (if (<= t_2 4e+293)
           (/ (+ (/ 1.0 (/ t (* z y))) x) t_1)
           (if (<= t_2 INFINITY) t_3 (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (1.0 + a) + ((b * y) / t);
	double t_2 = (((z * y) / t) + x) / t_1;
	double t_3 = (y / fma(b, y, fma(a, t, t))) * z;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -5e-324) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (z / b) + (((x / b) * t) / y);
	} else if (t_2 <= 4e+293) {
		tmp = ((1.0 / (t / (z * y))) + x) / t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t))
	t_2 = Float64(Float64(Float64(Float64(z * y) / t) + x) / t_1)
	t_3 = Float64(Float64(y / fma(b, y, fma(a, t, t))) * z)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -5e-324)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(Float64(Float64(x / b) * t) / y));
	elseif (t_2 <= 4e+293)
		tmp = Float64(Float64(Float64(1.0 / Float64(t / Float64(z * y))) + x) / t_1);
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -5e-324], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(z / b), $MachinePrecision] + N[(N[(N[(x / b), $MachinePrecision] * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+293], N[(N[(N[(1.0 / N[(t / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[(z / b), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 3.9999999999999997e293 < (/.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 42.5%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 87.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-324

   1. Initial program 99.8%

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

   if -4.94066e-324 < (/.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 46.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 80.8%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999997e293

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 98.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\frac{x}{b} \cdot t}{y}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{1}{\frac{t}{z \cdot y}} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ t_2 := \frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\frac{x}{b} \cdot t}{y}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (/ (* z y) t) x) (+ (+ 1.0 a) (/ (* b y) t))))
        (t_2 (* (/ y (fma b y (fma a t t))) z)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -5e-324)
       t_1
       (if (<= t_1 0.0)
         (+ (/ z b) (/ (* (/ x b) t) y))
         (if (<= t_1 4e+293) t_1 (if (<= t_1 INFINITY) t_2 (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * y) / t) + x) / ((1.0 + a) + ((b * y) / t));
	double t_2 = (y / fma(b, y, fma(a, t, t))) * z;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -5e-324) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (z / b) + (((x / b) * t) / y);
	} else if (t_1 <= 4e+293) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t)))
	t_2 = Float64(Float64(y / fma(b, y, fma(a, t, t))) * z)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -5e-324)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(Float64(Float64(x / b) * t) / y));
	elseif (t_1 <= 4e+293)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -5e-324], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(z / b), $MachinePrecision] + N[(N[(N[(x / b), $MachinePrecision] * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+293], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$2, N[(z / b), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 3.9999999999999997e293 < (/.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 42.5%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 87.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-324 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999997e293

   1. Initial program 99.1%

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

   if -4.94066e-324 < (/.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 46.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 80.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\frac{x}{b} \cdot t}{y}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t} + x\\ t_2 := \frac{t\_1}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ t_3 := \frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{t\_1}{1 + a}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\frac{x}{b} \cdot t}{y}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* z y) t) x))
        (t_2 (/ t_1 (+ (+ 1.0 a) (/ (* b y) t))))
        (t_3 (* (/ y (fma b y (fma a t t))) z)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -5e-324)
       (/ t_1 (+ 1.0 a))
       (if (<= t_2 0.0)
         (+ (/ z b) (/ (* (/ x b) t) y))
         (if (<= t_2 4e+293)
           (/ (fma z (/ y t) x) (+ 1.0 a))
           (if (<= t_2 INFINITY) t_3 (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((z * y) / t) + x;
	double t_2 = t_1 / ((1.0 + a) + ((b * y) / t));
	double t_3 = (y / fma(b, y, fma(a, t, t))) * z;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -5e-324) {
		tmp = t_1 / (1.0 + a);
	} else if (t_2 <= 0.0) {
		tmp = (z / b) + (((x / b) * t) / y);
	} else if (t_2 <= 4e+293) {
		tmp = fma(z, (y / t), x) / (1.0 + a);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(z * y) / t) + x)
	t_2 = Float64(t_1 / Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t)))
	t_3 = Float64(Float64(y / fma(b, y, fma(a, t, t))) * z)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -5e-324)
		tmp = Float64(t_1 / Float64(1.0 + a));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(z / b) + Float64(Float64(Float64(x / b) * t) / y));
	elseif (t_2 <= 4e+293)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(1.0 + a));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -5e-324], N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(z / b), $MachinePrecision] + N[(N[(N[(x / b), $MachinePrecision] * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+293], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[(z / b), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 3.9999999999999997e293 < (/.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 42.5%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 87.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-324

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

      1. lower-+.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if -4.94066e-324 < (/.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 46.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 80.8%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999997e293

   1. Initial program 98.7%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 71.6

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

   5. Applied rewrites 71.6%

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

   7. Applied rewrites 74.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 0:\\ \;\;\;\;\frac{z}{b} + \frac{\frac{x}{b} \cdot t}{y}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot y}{t} + x\\ t_2 := \frac{t\_1}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ t_3 := \frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{t\_1}{1 + a}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (/ (* z y) t) x))
        (t_2 (/ t_1 (+ (+ 1.0 a) (/ (* b y) t))))
        (t_3 (* (/ y (fma b y (fma a t t))) z)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -5e-324)
       (/ t_1 (+ 1.0 a))
       (if (<= t_2 0.0)
         (/ (fma t (/ x y) z) b)
         (if (<= t_2 4e+293)
           (/ (fma z (/ y t) x) (+ 1.0 a))
           (if (<= t_2 INFINITY) t_3 (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((z * y) / t) + x;
	double t_2 = t_1 / ((1.0 + a) + ((b * y) / t));
	double t_3 = (y / fma(b, y, fma(a, t, t))) * z;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -5e-324) {
		tmp = t_1 / (1.0 + a);
	} else if (t_2 <= 0.0) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t_2 <= 4e+293) {
		tmp = fma(z, (y / t), x) / (1.0 + a);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(z * y) / t) + x)
	t_2 = Float64(t_1 / Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t)))
	t_3 = Float64(Float64(y / fma(b, y, fma(a, t, t))) * z)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -5e-324)
		tmp = Float64(t_1 / Float64(1.0 + a));
	elseif (t_2 <= 0.0)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t_2 <= 4e+293)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(1.0 + a));
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -5e-324], N[(t$95$1 / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$2, 4e+293], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$3, N[(z / b), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 3.9999999999999997e293 < (/.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 42.5%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 87.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-324

   1. Initial program 99.8%

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

   3. Taylor expanded in b around 0

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

      1. lower-+.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if -4.94066e-324 < (/.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 46.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 72.5%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999997e293

   1. Initial program 98.7%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 71.6

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

   5. Applied rewrites 71.6%

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

   7. Applied rewrites 74.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{\frac{z \cdot y}{t} + x}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}\\ t_2 := \frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}}\\ t_3 := \frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma z (/ y t) x) (+ 1.0 a)))
        (t_2 (/ (+ (/ (* z y) t) x) (+ (+ 1.0 a) (/ (* b y) t))))
        (t_3 (* (/ y (fma b y (fma a t t))) z)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -5e-324)
       t_1
       (if (<= t_2 0.0)
         (/ (fma t (/ x y) z) b)
         (if (<= t_2 4e+293) t_1 (if (<= t_2 INFINITY) t_3 (/ z b))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (y / t), x) / (1.0 + a);
	double t_2 = (((z * y) / t) + x) / ((1.0 + a) + ((b * y) / t));
	double t_3 = (y / fma(b, y, fma(a, t, t))) * z;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -5e-324) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma(t, (x / y), z) / b;
	} else if (t_2 <= 4e+293) {
		tmp = t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(z, Float64(y / t), x) / Float64(1.0 + a))
	t_2 = Float64(Float64(Float64(Float64(z * y) / t) + x) / Float64(Float64(1.0 + a) + Float64(Float64(b * y) / t)))
	t_3 = Float64(Float64(y / fma(b, y, fma(a, t, t))) * z)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -5e-324)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (t_2 <= 4e+293)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(1.0 + a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + N[(N[(b * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -5e-324], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$2, 4e+293], t$95$1, If[LessEqual[t$95$2, Infinity], t$95$3, N[(z / b), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 3.9999999999999997e293 < (/.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 42.5%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 87.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -4.94066e-324 or -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 3.9999999999999997e293

   1. Initial program 99.1%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 73.1

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

   5. Applied rewrites 73.1%

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

   7. Applied rewrites 75.5%

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

   if -4.94066e-324 < (/.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 46.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 72.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -\infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{1 + a}\\ \mathbf{elif}\;\frac{\frac{z \cdot y}{t} + x}{\left(1 + a\right) + \frac{b \cdot y}{t}} \leq \infty:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{+58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;y \leq 9.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\frac{b}{t}, y, 1 + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4.9e+58)
   (/ (fma t (/ x y) z) b)
   (if (<= y 9.4e-7)
     (/ x (fma (/ b t) y (+ 1.0 a)))
     (* (/ y (fma b y (fma a t t))) z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.9e+58) {
		tmp = fma(t, (x / y), z) / b;
	} else if (y <= 9.4e-7) {
		tmp = x / fma((b / t), y, (1.0 + a));
	} else {
		tmp = (y / fma(b, y, fma(a, t, t))) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4.9e+58)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (y <= 9.4e-7)
		tmp = Float64(x / fma(Float64(b / t), y, Float64(1.0 + a)));
	else
		tmp = Float64(Float64(y / fma(b, y, fma(a, t, t))) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4.9e+58], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[y, 9.4e-7], N[(x / N[(N[(b / t), $MachinePrecision] * y + N[(1.0 + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.90000000000000018e58

   1. Initial program 51.3%

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

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 68.3%

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

   if -4.90000000000000018e58 < y < 9.4e-7

   1. Initial program 93.1%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]
   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. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. distribute-lft-neg-out N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-+.f64 70.9

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

   5. Applied rewrites 70.9%

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

   if 9.4e-7 < y

   1. Initial program 57.9%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 64.4%

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

Alternative 7: 59.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;y \leq 9.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b, y, \mathsf{fma}\left(a, t, t\right)\right)} \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -9e+36)
   (/ (fma t (/ x y) z) b)
   (if (<= y 9.4e-7) (/ x (+ 1.0 a)) (* (/ y (fma b y (fma a t t))) z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -9e+36) {
		tmp = fma(t, (x / y), z) / b;
	} else if (y <= 9.4e-7) {
		tmp = x / (1.0 + a);
	} else {
		tmp = (y / fma(b, y, fma(a, t, t))) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -9e+36)
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	elseif (y <= 9.4e-7)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = Float64(Float64(y / fma(b, y, fma(a, t, t))) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -9e+36], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[y, 9.4e-7], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(b * y + N[(a * t + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.99999999999999994e36

   1. Initial program 55.6%

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

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 66.5%

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

   if -8.99999999999999994e36 < y < 9.4e-7

   1. Initial program 94.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if 9.4e-7 < y

   1. Initial program 57.9%

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

   3. Taylor expanded in z around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 64.4%

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

Alternative 8: 61.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -9 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1050000:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -9e+36) t_1 (if (<= y 1050000.0) (/ x (+ 1.0 a)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, (x / y), z) / b;
	double tmp;
	if (y <= -9e+36) {
		tmp = t_1;
	} else if (y <= 1050000.0) {
		tmp = x / (1.0 + a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (y <= -9e+36)
		tmp = t_1;
	elseif (y <= 1050000.0)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -9e+36], t$95$1, If[LessEqual[y, 1050000.0], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.99999999999999994e36 or 1.05e6 < y

   1. Initial program 56.3%

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

   3. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

   5. Applied rewrites 48.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 63.7%

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

   if -8.99999999999999994e36 < y < 1.05e6

   1. Initial program 94.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 65.0

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

   5. Applied rewrites 65.0%

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

Alternative 9: 56.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{\frac{1 + a}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.5e+47)
   (/ z b)
   (if (<= y 2.8e+29) (/ 1.0 (/ (+ 1.0 a) x)) (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.5e+47) {
		tmp = z / b;
	} else if (y <= 2.8e+29) {
		tmp = 1.0 / ((1.0 + a) / x);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-3.5d+47)) then
        tmp = z / b
    else if (y <= 2.8d+29) then
        tmp = 1.0d0 / ((1.0d0 + a) / x)
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.5e+47) {
		tmp = z / b;
	} else if (y <= 2.8e+29) {
		tmp = 1.0 / ((1.0 + a) / x);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -3.5e+47:
		tmp = z / b
	elif y <= 2.8e+29:
		tmp = 1.0 / ((1.0 + a) / x)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3.5e+47)
		tmp = Float64(z / b);
	elseif (y <= 2.8e+29)
		tmp = Float64(1.0 / Float64(Float64(1.0 + a) / x));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -3.5e+47)
		tmp = z / b;
	elseif (y <= 2.8e+29)
		tmp = 1.0 / ((1.0 + a) / x);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3.5e+47], N[(z / b), $MachinePrecision], If[LessEqual[y, 2.8e+29], N[(1.0 / N[(N[(1.0 + a), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.50000000000000015e47 or 2.8e29 < y

   1. Initial program 54.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 59.1

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

   5. Applied rewrites 59.1%

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

   if -3.50000000000000015e47 < y < 2.8e29

   1. Initial program 92.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 92.4

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 87.5

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 87.5

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-/.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. associate-/l* N/A

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

      21. *-commutative N/A

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

      22. lower-fma.f64 N/A

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

      23. lower-/.f64 84.6

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 62.7

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

   7. Applied rewrites 62.7%

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

Alternative 10: 41.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-252}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-110}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.5e+47)
   (/ z b)
   (if (<= y -5e-252) (/ x a) (if (<= y 3e-110) (/ x 1.0) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.5e+47) {
		tmp = z / b;
	} else if (y <= -5e-252) {
		tmp = x / a;
	} else if (y <= 3e-110) {
		tmp = x / 1.0;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-3.5d+47)) then
        tmp = z / b
    else if (y <= (-5d-252)) then
        tmp = x / a
    else if (y <= 3d-110) then
        tmp = x / 1.0d0
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.5e+47) {
		tmp = z / b;
	} else if (y <= -5e-252) {
		tmp = x / a;
	} else if (y <= 3e-110) {
		tmp = x / 1.0;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -3.5e+47:
		tmp = z / b
	elif y <= -5e-252:
		tmp = x / a
	elif y <= 3e-110:
		tmp = x / 1.0
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3.5e+47)
		tmp = Float64(z / b);
	elseif (y <= -5e-252)
		tmp = Float64(x / a);
	elseif (y <= 3e-110)
		tmp = Float64(x / 1.0);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -3.5e+47)
		tmp = z / b;
	elseif (y <= -5e-252)
		tmp = x / a;
	elseif (y <= 3e-110)
		tmp = x / 1.0;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3.5e+47], N[(z / b), $MachinePrecision], If[LessEqual[y, -5e-252], N[(x / a), $MachinePrecision], If[LessEqual[y, 3e-110], N[(x / 1.0), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.50000000000000015e47 or 2.99999999999999986e-110 < y

   1. Initial program 60.6%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 53.5

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

   5. Applied rewrites 53.5%

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

   if -3.50000000000000015e47 < y < -5.00000000000000008e-252

   1. Initial program 89.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]
   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. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. distribute-lft-neg-out N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-+.f64 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 38.1%

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

   if -5.00000000000000008e-252 < y < 2.99999999999999986e-110

   1. Initial program 97.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 49.0%

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

Alternative 11: 56.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{1 + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.5e+47) (/ z b) (if (<= y 2.8e+29) (/ x (+ 1.0 a)) (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.5e+47) {
		tmp = z / b;
	} else if (y <= 2.8e+29) {
		tmp = x / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-3.5d+47)) then
        tmp = z / b
    else if (y <= 2.8d+29) then
        tmp = x / (1.0d0 + a)
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.5e+47) {
		tmp = z / b;
	} else if (y <= 2.8e+29) {
		tmp = x / (1.0 + a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -3.5e+47:
		tmp = z / b
	elif y <= 2.8e+29:
		tmp = x / (1.0 + a)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3.5e+47)
		tmp = Float64(z / b);
	elseif (y <= 2.8e+29)
		tmp = Float64(x / Float64(1.0 + a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -3.5e+47)
		tmp = z / b;
	elseif (y <= 2.8e+29)
		tmp = x / (1.0 + a);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3.5e+47], N[(z / b), $MachinePrecision], If[LessEqual[y, 2.8e+29], N[(x / N[(1.0 + a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.50000000000000015e47 or 2.8e29 < y

   1. Initial program 54.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 59.1

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

   5. Applied rewrites 59.1%

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

   if -3.50000000000000015e47 < y < 2.8e29

   1. Initial program 92.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 62.6

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

   5. Applied rewrites 62.6%

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

Alternative 12: 40.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{elif}\;a \leq 16500:\\ \;\;\;\;\mathsf{fma}\left(-x, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.0) (/ x a) (if (<= a 16500.0) (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) {
		tmp = x / a;
	} else if (a <= 16500.0) {
		tmp = fma(-x, a, x);
	} else {
		tmp = x / a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.0)
		tmp = Float64(x / a);
	elseif (a <= 16500.0)
		tmp = fma(Float64(-x), a, x);
	else
		tmp = Float64(x / a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.0], N[(x / a), $MachinePrecision], If[LessEqual[a, 16500.0], N[((-x) * a + x), $MachinePrecision], N[(x / a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1 or 16500 < a

   1. Initial program 74.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x}{1 + \left(a + \frac{b \cdot y}{t}\right)}} \]
   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. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. distribute-lft-neg-out N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-+.f64 50.7

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

   5. Applied rewrites 50.7%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 44.4%

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

   if -1 < a < 16500

   1. Initial program 75.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 35.5

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

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 35.1%

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

Alternative 13: 19.1% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 40.5

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

5. Applied rewrites 40.5%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 19.0%

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

Alternative 14: 4.1% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 40.5

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

5. Applied rewrites 40.5%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 19.0%

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

9. Taylor expanded in a around inf

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

11. Applied rewrites 3.4%

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

Developer Target 1: 79.4% 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 2024276 
(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))))