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

Percentage Accurate: 74.8% → 88.5%

Time: 13.1s

Alternatives: 15

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.8% 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: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \mathsf{fma}\left(y, \frac{b}{t}, a\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;z \cdot \left(\frac{x}{\mathsf{fma}\left(z, t\_2, z\right)} + \frac{y}{\mathsf{fma}\left(t, t\_2, t\right)}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0))))
        (t_2 (fma y (/ b t) a)))
   (if (<= t_1 (- INFINITY))
     (* z (+ (/ x (fma z t_2 z)) (/ y (fma t t_2 t))))
     (if (<= t_1 1e+304) t_1 (/ (fma t (/ x y) z) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double t_2 = fma(y, (b / t), a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = z * ((x / fma(z, t_2, z)) + (y / fma(t, t_2, t)));
	} else if (t_1 <= 1e+304) {
		tmp = t_1;
	} else {
		tmp = fma(t, (x / y), z) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_2 = fma(y, Float64(b / t), a)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(z * Float64(Float64(x / fma(z, t_2, z)) + Float64(y / fma(t, t_2, t))));
	elseif (t_1 <= 1e+304)
		tmp = t_1;
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(z * N[(N[(x / N[(z * t$95$2 + z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * t$95$2 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+304], t$95$1, N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 26.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

   5. Applied rewrites 88.9%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999994e303

   1. Initial program 91.6%

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

   if 9.9999999999999994e303 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 14.4%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 18.8

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

   5. Applied rewrites 18.8%

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

   7. Applied rewrites 46.7%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 89.5%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;z \cdot \left(\frac{x}{\mathsf{fma}\left(z, \mathsf{fma}\left(y, \frac{b}{t}, a\right), z\right)} + \frac{y}{\mathsf{fma}\left(t, \mathsf{fma}\left(y, \frac{b}{t}, a\right), t\right)}\right)\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+304}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_3 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t, x, y \cdot z\right)}{y}}{b}\\ \mathbf{elif}\;t\_2 \leq 10^{+304}:\\ \;\;\;\;\frac{t\_1}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ (/ (* y b) t) (+ a 1.0))))
        (t_3 (/ (fma t (/ x y) z) b)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -1e-288)
       (/ (fma z (/ y t) x) (+ a 1.0))
       (if (<= t_2 0.0)
         (/ (/ (fma t x (* y z)) y) b)
         (if (<= t_2 1e+304) (/ t_1 (+ a 1.0)) t_3))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / (((y * b) / t) + (a + 1.0));
	double t_3 = fma(t, (x / y), z) / b;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -1e-288) {
		tmp = fma(z, (y / t), x) / (a + 1.0);
	} else if (t_2 <= 0.0) {
		tmp = (fma(t, x, (y * z)) / y) / b;
	} else if (t_2 <= 1e+304) {
		tmp = t_1 / (a + 1.0);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_3 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -1e-288)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(a + 1.0));
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(fma(t, x, Float64(y * z)) / y) / b);
	elseif (t_2 <= 1e+304)
		tmp = Float64(t_1 / Float64(a + 1.0));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -1e-288], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[(t * x + N[(y * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$2, 1e+304], N[(t$95$1 / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 9.9999999999999994e303 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 19.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 21.8

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

   5. Applied rewrites 21.8%

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

   7. Applied rewrites 48.1%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 78.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -1.00000000000000006e-288

   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-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 78.2

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

   5. Applied rewrites 78.2%

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

   if -1.00000000000000006e-288 < (/.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 55.5%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 54.3

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

   5. Applied rewrites 54.3%

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

   7. Applied rewrites 77.2%

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

   if -0.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999994e303

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 72.7

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

   5. Applied rewrites 72.7%

      \[\leadsto \frac{x + \frac{y \cdot z}{t}}{\color{blue}{1 + a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -1 \cdot 10^{-288}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 0:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(t, x, y \cdot z\right)}{y}}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+304}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_2 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+304}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0))))
        (t_2 (/ (fma t (/ x y) z) b)))
   (if (<= t_1 (- INFINITY)) t_2 (if (<= t_1 1e+304) t_1 t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0));
	double t_2 = fma(t, (x / y), z) / b;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 1e+304) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_2 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 1e+304)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 1e+304], t$95$1, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 9.9999999999999994e303 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 19.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 21.8

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

   5. Applied rewrites 21.8%

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

   7. Applied rewrites 48.1%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 78.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999994e303

   1. Initial program 91.6%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+304}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot z}{t}\\ t_2 := \frac{t\_1}{\frac{y \cdot b}{t} + \left(a + 1\right)}\\ t_3 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{+304}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y z) t)))
        (t_2 (/ t_1 (+ (/ (* y b) t) (+ a 1.0))))
        (t_3 (/ (fma t (/ x y) z) b)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 1e+304) (/ t_1 (fma b (/ y t) (+ a 1.0))) t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + ((y * z) / t);
	double t_2 = t_1 / (((y * b) / t) + (a + 1.0));
	double t_3 = fma(t, (x / y), z) / b;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 1e+304) {
		tmp = t_1 / fma(b, (y / t), (a + 1.0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(y * z) / t))
	t_2 = Float64(t_1 / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0)))
	t_3 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 1e+304)
		tmp = Float64(t_1 / fma(b, Float64(y / t), Float64(a + 1.0)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, 1e+304], N[(t$95$1 / N[(b * N[(y / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < -inf.0 or 9.9999999999999994e303 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 19.6%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 21.8

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

   5. Applied rewrites 21.8%

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

   7. Applied rewrites 48.1%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 78.5%

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

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999994e303

   1. Initial program 91.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 91.2

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

   4. Applied rewrites 91.2%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+304}:\\ \;\;\;\;\frac{x + \frac{y \cdot z}{t}}{\mathsf{fma}\left(b, \frac{y}{t}, a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+304}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (/ (+ x (/ (* y z) t)) (+ (/ (* y b) t) (+ a 1.0))) 1e+304)
   (/ (fma y (/ z t) x) (fma y (/ b t) (+ a 1.0)))
   (/ (fma t (/ x y) z) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((x + ((y * z) / t)) / (((y * b) / t) + (a + 1.0))) <= 1e+304) {
		tmp = fma(y, (z / t), x) / fma(y, (b / t), (a + 1.0));
	} else {
		tmp = fma(t, (x / y), z) / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(x + Float64(Float64(y * z) / t)) / Float64(Float64(Float64(y * b) / t) + Float64(a + 1.0))) <= 1e+304)
		tmp = Float64(fma(y, Float64(z / t), x) / fma(y, Float64(b / t), Float64(a + 1.0)));
	else
		tmp = Float64(fma(t, Float64(x / y), z) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(x + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+304], N[(N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision] / N[(y * N[(b / t), $MachinePrecision] + N[(a + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t))) < 9.9999999999999994e303

   1. Initial program 83.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 81.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 80.3

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

   4. Applied rewrites 80.3%

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

   if 9.9999999999999994e303 < (/.f64 (+.f64 x (/.f64 (*.f64 y z) t)) (+.f64 (+.f64 a #s(literal 1 binary64)) (/.f64 (*.f64 y b) t)))

   1. Initial program 14.4%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 18.8

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

   5. Applied rewrites 18.8%

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

   7. Applied rewrites 46.7%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 89.5%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z}{t}}{\frac{y \cdot b}{t} + \left(a + 1\right)} \leq 10^{+304}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{z}{t}, x\right)}{\mathsf{fma}\left(y, \frac{b}{t}, a + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -0.0034:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -0.0034)
     t_1
     (if (<= y 7.2e+109) (/ (fma z (/ y t) x) (+ a 1.0)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, (x / y), z) / b;
	double tmp;
	if (y <= -0.0034) {
		tmp = t_1;
	} else if (y <= 7.2e+109) {
		tmp = fma(z, (y / t), x) / (a + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (y <= -0.0034)
		tmp = t_1;
	elseif (y <= 7.2e+109)
		tmp = Float64(fma(z, Float64(y / t), x) / Float64(a + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -0.0034], t$95$1, If[LessEqual[y, 7.2e+109], N[(N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision] / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00339999999999999981 or 7.2e109 < y

   1. Initial program 51.1%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 31.5

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

   5. Applied rewrites 31.5%

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

   7. Applied rewrites 52.2%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 70.1%

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

   if -0.00339999999999999981 < y < 7.2e109

   1. Initial program 92.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 75.2

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

   5. Applied rewrites 75.2%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0034:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{y}{t}, x\right)}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -2.6 \cdot 10^{-34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+105}:\\ \;\;\;\;\frac{x}{1 + \mathsf{fma}\left(y, \frac{b}{t}, a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -2.6e-34)
     t_1
     (if (<= y 9.5e+105) (/ x (+ 1.0 (fma y (/ b t) a))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, (x / y), z) / b;
	double tmp;
	if (y <= -2.6e-34) {
		tmp = t_1;
	} else if (y <= 9.5e+105) {
		tmp = x / (1.0 + fma(y, (b / t), a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (y <= -2.6e-34)
		tmp = t_1;
	elseif (y <= 9.5e+105)
		tmp = Float64(x / Float64(1.0 + fma(y, Float64(b / t), a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -2.6e-34], t$95$1, If[LessEqual[y, 9.5e+105], N[(x / N[(1.0 + N[(y * N[(b / t), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5999999999999999e-34 or 9.4999999999999995e105 < y

   1. Initial program 53.9%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 31.2

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

   5. Applied rewrites 31.2%

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

   7. Applied rewrites 50.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 67.1%

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

   if -2.5999999999999999e-34 < y < 9.4999999999999995e105

   1. Initial program 92.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 71.5

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

   5. Applied rewrites 71.5%

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

Alternative 8: 59.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (fma t (/ x y) z) b)))
   (if (<= y -8.5e-39) t_1 (if (<= y 2.4e-95) (/ x (+ a 1.0)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(t, (x / y), z) / b;
	double tmp;
	if (y <= -8.5e-39) {
		tmp = t_1;
	} else if (y <= 2.4e-95) {
		tmp = x / (a + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(t, Float64(x / y), z) / b)
	tmp = 0.0
	if (y <= -8.5e-39)
		tmp = t_1;
	elseif (y <= 2.4e-95)
		tmp = Float64(x / Float64(a + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t * N[(x / y), $MachinePrecision] + z), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[y, -8.5e-39], t$95$1, If[LessEqual[y, 2.4e-95], N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5000000000000005e-39 or 2.4e-95 < y

   1. Initial program 57.7%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 33.1

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

   5. Applied rewrites 33.1%

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

   7. Applied rewrites 50.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 62.8%

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

   if -8.5000000000000005e-39 < y < 2.4e-95

   1. Initial program 96.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 69.2

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

   5. Applied rewrites 69.2%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-95}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, \frac{x}{y}, z\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+66}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, z, x \cdot t\right)}{y \cdot b}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.22e+66)
   (/ z b)
   (if (<= y -8.5e-39)
     (/ (fma y z (* x t)) (* y b))
     (if (<= y 6.8e+105) (/ x (+ a 1.0)) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.22e+66) {
		tmp = z / b;
	} else if (y <= -8.5e-39) {
		tmp = fma(y, z, (x * t)) / (y * b);
	} else if (y <= 6.8e+105) {
		tmp = x / (a + 1.0);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.22e+66)
		tmp = Float64(z / b);
	elseif (y <= -8.5e-39)
		tmp = Float64(fma(y, z, Float64(x * t)) / Float64(y * b));
	elseif (y <= 6.8e+105)
		tmp = Float64(x / Float64(a + 1.0));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.22e+66], N[(z / b), $MachinePrecision], If[LessEqual[y, -8.5e-39], N[(N[(y * z + N[(x * t), $MachinePrecision]), $MachinePrecision] / N[(y * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.8e+105], N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.21999999999999993e66 or 6.7999999999999999e105 < y

   1. Initial program 50.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 61.0

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

   5. Applied rewrites 61.0%

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

   if -1.21999999999999993e66 < y < -8.5000000000000005e-39

   1. Initial program 66.8%

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

   3. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 46.7

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

   5. Applied rewrites 46.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 56.6%

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

   if -8.5000000000000005e-39 < y < 6.7999999999999999e105

   1. Initial program 92.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 63.9

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

   5. Applied rewrites 63.9%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+66}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, z, x \cdot t\right)}{y \cdot b}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-39}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -5e-39)
   (/ z b)
   (if (<= y -5e-308) (/ x 1.0) (if (<= y 1.65e-178) (/ x a) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5e-39) {
		tmp = z / b;
	} else if (y <= -5e-308) {
		tmp = x / 1.0;
	} else if (y <= 1.65e-178) {
		tmp = x / a;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-5d-39)) then
        tmp = z / b
    else if (y <= (-5d-308)) then
        tmp = x / 1.0d0
    else if (y <= 1.65d-178) then
        tmp = x / a
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5e-39) {
		tmp = z / b;
	} else if (y <= -5e-308) {
		tmp = x / 1.0;
	} else if (y <= 1.65e-178) {
		tmp = x / a;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -5e-39:
		tmp = z / b
	elif y <= -5e-308:
		tmp = x / 1.0
	elif y <= 1.65e-178:
		tmp = x / a
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -5e-39)
		tmp = Float64(z / b);
	elseif (y <= -5e-308)
		tmp = Float64(x / 1.0);
	elseif (y <= 1.65e-178)
		tmp = Float64(x / a);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -5e-39)
		tmp = z / b;
	elseif (y <= -5e-308)
		tmp = x / 1.0;
	elseif (y <= 1.65e-178)
		tmp = x / a;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -5e-39], N[(z / b), $MachinePrecision], If[LessEqual[y, -5e-308], N[(x / 1.0), $MachinePrecision], If[LessEqual[y, 1.65e-178], N[(x / a), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.9999999999999998e-39 or 1.6500000000000001e-178 < y

   1. Initial program 61.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 49.4

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

   5. Applied rewrites 49.4%

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

   if -4.9999999999999998e-39 < y < -4.99999999999999955e-308

   1. Initial program 95.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 87.8

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

   4. Applied rewrites 87.8%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 66.4

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

   7. Applied rewrites 66.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 40.4%

       \[\leadsto \frac{x}{1} \]

   if -4.99999999999999955e-308 < y < 1.6500000000000001e-178

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 88.2

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 57.4%

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

Alternative 11: 39.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-39}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-308}:\\ \;\;\;\;x - x \cdot a\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -5e-39)
   (/ z b)
   (if (<= y -5e-308) (- x (* x a)) (if (<= y 1.65e-178) (/ x a) (/ z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5e-39) {
		tmp = z / b;
	} else if (y <= -5e-308) {
		tmp = x - (x * a);
	} else if (y <= 1.65e-178) {
		tmp = x / a;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-5d-39)) then
        tmp = z / b
    else if (y <= (-5d-308)) then
        tmp = x - (x * a)
    else if (y <= 1.65d-178) then
        tmp = x / a
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5e-39) {
		tmp = z / b;
	} else if (y <= -5e-308) {
		tmp = x - (x * a);
	} else if (y <= 1.65e-178) {
		tmp = x / a;
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -5e-39:
		tmp = z / b
	elif y <= -5e-308:
		tmp = x - (x * a)
	elif y <= 1.65e-178:
		tmp = x / a
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -5e-39)
		tmp = Float64(z / b);
	elseif (y <= -5e-308)
		tmp = Float64(x - Float64(x * a));
	elseif (y <= 1.65e-178)
		tmp = Float64(x / a);
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -5e-39)
		tmp = z / b;
	elseif (y <= -5e-308)
		tmp = x - (x * a);
	elseif (y <= 1.65e-178)
		tmp = x / a;
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -5e-39], N[(z / b), $MachinePrecision], If[LessEqual[y, -5e-308], N[(x - N[(x * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.65e-178], N[(x / a), $MachinePrecision], N[(z / b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.9999999999999998e-39 or 1.6500000000000001e-178 < y

   1. Initial program 61.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 49.4

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

   5. Applied rewrites 49.4%

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

   if -4.9999999999999998e-39 < y < -4.99999999999999955e-308

   1. Initial program 95.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 87.8

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

   4. Applied rewrites 87.8%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 66.4

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

   7. Applied rewrites 66.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 39.5%

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

   if -4.99999999999999955e-308 < y < 1.6500000000000001e-178

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 88.2

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in a around inf

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

   10. Applied rewrites 57.4%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-39}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-308}:\\ \;\;\;\;x - x \cdot a\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-178}:\\ \;\;\;\;\frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-39}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -9e-39) (/ z b) (if (<= y 6.8e+105) (/ x (+ a 1.0)) (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -9e-39) {
		tmp = z / b;
	} else if (y <= 6.8e+105) {
		tmp = x / (a + 1.0);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-9d-39)) then
        tmp = z / b
    else if (y <= 6.8d+105) then
        tmp = x / (a + 1.0d0)
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -9e-39) {
		tmp = z / b;
	} else if (y <= 6.8e+105) {
		tmp = x / (a + 1.0);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -9e-39:
		tmp = z / b
	elif y <= 6.8e+105:
		tmp = x / (a + 1.0)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -9e-39)
		tmp = Float64(z / b);
	elseif (y <= 6.8e+105)
		tmp = Float64(x / Float64(a + 1.0));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -9e-39)
		tmp = z / b;
	elseif (y <= 6.8e+105)
		tmp = x / (a + 1.0);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -9e-39], N[(z / b), $MachinePrecision], If[LessEqual[y, 6.8e+105], N[(x / N[(a + 1.0), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.0000000000000002e-39 or 6.7999999999999999e105 < y

   1. Initial program 54.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 55.4

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

   5. Applied rewrites 55.4%

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

   if -9.0000000000000002e-39 < y < 6.7999999999999999e105

   1. Initial program 92.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 63.9

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

   5. Applied rewrites 63.9%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{-39}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{x}{a + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-39}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-108}:\\ \;\;\;\;x - x \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -5e-39) (/ z b) (if (<= y 5.2e-108) (- x (* x a)) (/ z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5e-39) {
		tmp = z / b;
	} else if (y <= 5.2e-108) {
		tmp = x - (x * a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-5d-39)) then
        tmp = z / b
    else if (y <= 5.2d-108) then
        tmp = x - (x * a)
    else
        tmp = z / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -5e-39) {
		tmp = z / b;
	} else if (y <= 5.2e-108) {
		tmp = x - (x * a);
	} else {
		tmp = z / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -5e-39:
		tmp = z / b
	elif y <= 5.2e-108:
		tmp = x - (x * a)
	else:
		tmp = z / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -5e-39)
		tmp = Float64(z / b);
	elseif (y <= 5.2e-108)
		tmp = Float64(x - Float64(x * a));
	else
		tmp = Float64(z / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -5e-39)
		tmp = z / b;
	elseif (y <= 5.2e-108)
		tmp = x - (x * a);
	else
		tmp = z / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -5e-39], N[(z / b), $MachinePrecision], If[LessEqual[y, 5.2e-108], N[(x - N[(x * a), $MachinePrecision]), $MachinePrecision], N[(z / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.9999999999999998e-39 or 5.19999999999999968e-108 < 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 y around inf

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

      1. lower-/.f64 52.0

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

   5. Applied rewrites 52.0%

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

   if -4.9999999999999998e-39 < y < 5.19999999999999968e-108

   1. Initial program 96.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 88.9

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

   4. Applied rewrites 88.9%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 68.9

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

   7. Applied rewrites 68.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 36.8%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-39}:\\ \;\;\;\;\frac{z}{b}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-108}:\\ \;\;\;\;x - x \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 19.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 72.6

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

4. Applied rewrites 72.6%

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

5. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 40.9

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

7. Applied rewrites 40.9%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 22.6%

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

11. Final simplification 22.6%

    \[\leadsto x - x \cdot a \]
12. Add Preprocessing

Alternative 15: 4.1% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 72.6

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

4. Applied rewrites 72.6%

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

5. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 40.9

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

7. Applied rewrites 40.9%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 22.6%

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

11. Taylor expanded in a around inf

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

13. Applied rewrites 4.8%

    \[\leadsto \left(-a\right) \cdot x \]

14. Final simplification 4.8%

    \[\leadsto -x \cdot a \]
15. Add Preprocessing

Developer Target 1: 79.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024216 
(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))))