AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1

Percentage Accurate: 60.6% → 89.0%

Time: 10.2s

Alternatives: 15

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

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 + y) * a)) - (y * b)) / ((x + t) + y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + y) * a)) - (y * b)) / ((x + t) + y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ t y) a) (* z (+ y x))) (* b y)) (+ (+ t x) y)))
        (t_2 (fma y (/ (- z b) (+ t y)) a)))
   (if (<= t_1 (- INFINITY)) t_2 (if (<= t_1 1e+294) t_1 t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / ((t + x) + y);
	double t_2 = fma(y, ((z - b) / (t + y)), a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 1e+294) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(t + y) * a) + Float64(z * Float64(y + x))) - Float64(b * y)) / Float64(Float64(t + x) + y))
	t_2 = fma(y, Float64(Float64(z - b) / Float64(t + y)), a)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 1e+294)
		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[(N[(N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(N[(z - b), $MachinePrecision] / N[(t + y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 1e+294], t$95$1, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0 or 1.00000000000000007e294 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 5.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. associate--l+ N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 10.3

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

   5. Applied rewrites 10.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 81.8%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.00000000000000007e294

   1. Initial program 99.7%

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

4. Final simplification 92.7%

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

Alternative 2: 78.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ t y) a) (* z (+ y x))) (* b y)) (+ (+ t x) y)))
        (t_2 (fma y (/ (- z b) (+ t y)) a)))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 1e+294)
       (fma 1.0 a (/ (fma x z (* (- z b) y)) (+ t (+ y x))))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / ((t + x) + y);
	double t_2 = fma(y, ((z - b) / (t + y)), a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 1e+294) {
		tmp = fma(1.0, a, (fma(x, z, ((z - b) * y)) / (t + (y + x))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(t + y) * a) + Float64(z * Float64(y + x))) - Float64(b * y)) / Float64(Float64(t + x) + y))
	t_2 = fma(y, Float64(Float64(z - b) / Float64(t + y)), a)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 1e+294)
		tmp = fma(1.0, a, Float64(fma(x, z, Float64(Float64(z - b) * y)) / Float64(t + Float64(y + x))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0 or 1.00000000000000007e294 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 5.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. associate--l+ N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 10.3

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

   5. Applied rewrites 10.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 81.8%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.00000000000000007e294

   1. Initial program 99.7%

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

   3. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. div-sub N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 81.7%

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

4. Final simplification 81.7%

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

Alternative 3: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z - b}{t + y}, a\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{-118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-137}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y + x, z, \left(t + y\right) \cdot a\right)}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (/ (- z b) (+ t y)) a)))
   (if (<= y -5e-118)
     t_1
     (if (<= y 6e-137) (/ (fma (+ y x) z (* (+ t y) a)) (+ (+ t x) y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, ((z - b) / (t + y)), a);
	double tmp;
	if (y <= -5e-118) {
		tmp = t_1;
	} else if (y <= 6e-137) {
		tmp = fma((y + x), z, ((t + y) * a)) / ((t + x) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(Float64(z - b) / Float64(t + y)), a)
	tmp = 0.0
	if (y <= -5e-118)
		tmp = t_1;
	elseif (y <= 6e-137)
		tmp = Float64(fma(Float64(y + x), z, Float64(Float64(t + y) * a)) / Float64(Float64(t + x) + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(N[(z - b), $MachinePrecision] / N[(t + y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]}, If[LessEqual[y, -5e-118], t$95$1, If[LessEqual[y, 6e-137], N[(N[(N[(y + x), $MachinePrecision] * z + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000015e-118 or 5.9999999999999996e-137 < y

   1. Initial program 53.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. associate--l+ N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 48.6

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

   5. Applied rewrites 48.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 81.7%

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

   if -5.00000000000000015e-118 < y < 5.9999999999999996e-137

   1. Initial program 83.6%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 75.4

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

   5. Applied rewrites 75.4%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-118}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{t + y}, a\right)\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-137}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y + x, z, \left(t + y\right) \cdot a\right)}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{t + y}, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z - b}{t + y}, a\right)\\ \mathbf{if}\;y \leq -4.3 \cdot 10^{-118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, z, a \cdot t\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (/ (- z b) (+ t y)) a)))
   (if (<= y -4.3e-118)
     t_1
     (if (<= y 3.6e-138) (/ (fma x z (* a t)) (+ t x)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, ((z - b) / (t + y)), a);
	double tmp;
	if (y <= -4.3e-118) {
		tmp = t_1;
	} else if (y <= 3.6e-138) {
		tmp = fma(x, z, (a * t)) / (t + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(Float64(z - b) / Float64(t + y)), a)
	tmp = 0.0
	if (y <= -4.3e-118)
		tmp = t_1;
	elseif (y <= 3.6e-138)
		tmp = Float64(fma(x, z, Float64(a * t)) / Float64(t + x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(N[(z - b), $MachinePrecision] / N[(t + y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]}, If[LessEqual[y, -4.3e-118], t$95$1, If[LessEqual[y, 3.6e-138], N[(N[(x * z + N[(a * t), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.30000000000000018e-118 or 3.60000000000000018e-138 < y

   1. Initial program 53.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. associate--l+ N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 48.6

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

   5. Applied rewrites 48.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 81.7%

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

   if -4.30000000000000018e-118 < y < 3.60000000000000018e-138

   1. Initial program 83.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 74.5

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

   5. Applied rewrites 74.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-118}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{t + y}, a\right)\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, z, a \cdot t\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{t + y}, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+252}:\\ \;\;\;\;z - \frac{b - a}{x} \cdot y\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{t + y}, a\right)\\ \mathbf{else}:\\ \;\;\;\;z - \left(-a\right) \cdot \frac{t + y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -4.4e+252)
   (- z (* (/ (- b a) x) y))
   (if (<= x 3.3e+31)
     (fma y (/ (- z b) (+ t y)) a)
     (- z (* (- a) (/ (+ t y) x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -4.4e+252) {
		tmp = z - (((b - a) / x) * y);
	} else if (x <= 3.3e+31) {
		tmp = fma(y, ((z - b) / (t + y)), a);
	} else {
		tmp = z - (-a * ((t + y) / x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -4.4e+252)
		tmp = Float64(z - Float64(Float64(Float64(b - a) / x) * y));
	elseif (x <= 3.3e+31)
		tmp = fma(y, Float64(Float64(z - b) / Float64(t + y)), a);
	else
		tmp = Float64(z - Float64(Float64(-a) * Float64(Float64(t + y) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -4.4e+252], N[(z - N[(N[(N[(b - a), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.3e+31], N[(y * N[(N[(z - b), $MachinePrecision] / N[(t + y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], N[(z - N[((-a) * N[(N[(t + y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4000000000000001e252

   1. Initial program 27.5%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 87.6%

      \[\leadsto z - \frac{b - a}{x} \cdot \color{blue}{y} \]

   if -4.4000000000000001e252 < x < 3.29999999999999992e31

   1. Initial program 63.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. associate--l+ N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 50.3

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.4%

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

   if 3.29999999999999992e31 < x

   1. Initial program 66.6%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in a around -inf

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

   8. Applied rewrites 66.5%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+252}:\\ \;\;\;\;z - \frac{b - a}{x} \cdot y\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{t + y}, a\right)\\ \mathbf{else}:\\ \;\;\;\;z - \left(-a\right) \cdot \frac{t + y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+252}:\\ \;\;\;\;z - \frac{b - a}{x} \cdot y\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{t + y}, a\right)\\ \mathbf{else}:\\ \;\;\;\;z - \frac{z - a}{x} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -4.4e+252)
   (- z (* (/ (- b a) x) y))
   (if (<= x 3.3e+31)
     (fma y (/ (- z b) (+ t y)) a)
     (- z (* (/ (- z a) x) t)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -4.4e+252) {
		tmp = z - (((b - a) / x) * y);
	} else if (x <= 3.3e+31) {
		tmp = fma(y, ((z - b) / (t + y)), a);
	} else {
		tmp = z - (((z - a) / x) * t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -4.4e+252)
		tmp = Float64(z - Float64(Float64(Float64(b - a) / x) * y));
	elseif (x <= 3.3e+31)
		tmp = fma(y, Float64(Float64(z - b) / Float64(t + y)), a);
	else
		tmp = Float64(z - Float64(Float64(Float64(z - a) / x) * t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -4.4e+252], N[(z - N[(N[(N[(b - a), $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.3e+31], N[(y * N[(N[(z - b), $MachinePrecision] / N[(t + y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], N[(z - N[(N[(N[(z - a), $MachinePrecision] / x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4000000000000001e252

   1. Initial program 27.5%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 87.6%

      \[\leadsto z - \frac{b - a}{x} \cdot \color{blue}{y} \]

   if -4.4000000000000001e252 < x < 3.29999999999999992e31

   1. Initial program 63.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. associate--l+ N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 50.3

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

   5. Applied rewrites 50.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.4%

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

   if 3.29999999999999992e31 < x

   1. Initial program 66.6%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 63.3%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+252}:\\ \;\;\;\;z - \frac{b - a}{x} \cdot y\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{t + y}, a\right)\\ \mathbf{else}:\\ \;\;\;\;z - \frac{z - a}{x} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - b}{t}, y, a\right)\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+121}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ (- z b) t) y a)))
   (if (<= t -1.3e+141) t_1 (if (<= t 3.7e+121) (- (+ a z) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(((z - b) / t), y, a);
	double tmp;
	if (t <= -1.3e+141) {
		tmp = t_1;
	} else if (t <= 3.7e+121) {
		tmp = (a + z) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(Float64(z - b) / t), y, a)
	tmp = 0.0
	if (t <= -1.3e+141)
		tmp = t_1;
	elseif (t <= 3.7e+121)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z - b), $MachinePrecision] / t), $MachinePrecision] * y + a), $MachinePrecision]}, If[LessEqual[t, -1.3e+141], t$95$1, If[LessEqual[t, 3.7e+121], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.3e141 or 3.70000000000000013e121 < t

   1. Initial program 49.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. associate--l+ N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 80.2%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 73.1%

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

   if -1.3e141 < t < 3.70000000000000013e121

   1. Initial program 69.2%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(a + z\right) - b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

      3. lower-+.f64 61.9

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   5. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - b}{t}, y, a\right)\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+121}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - b}{t}, y, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -1.36 \cdot 10^{-98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-149}:\\ \;\;\;\;\frac{z}{t + x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -1.36e-98) t_1 (if (<= y 9e-149) (* (/ z (+ t x)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double tmp;
	if (y <= -1.36e-98) {
		tmp = t_1;
	} else if (y <= 9e-149) {
		tmp = (z / (t + x)) * x;
	} 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 = (a + z) - b
    if (y <= (-1.36d-98)) then
        tmp = t_1
    else if (y <= 9d-149) then
        tmp = (z / (t + x)) * x
    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 = (a + z) - b;
	double tmp;
	if (y <= -1.36e-98) {
		tmp = t_1;
	} else if (y <= 9e-149) {
		tmp = (z / (t + x)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a + z) - b
	tmp = 0
	if y <= -1.36e-98:
		tmp = t_1
	elif y <= 9e-149:
		tmp = (z / (t + x)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -1.36e-98)
		tmp = t_1;
	elseif (y <= 9e-149)
		tmp = Float64(Float64(z / Float64(t + x)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a + z) - b;
	tmp = 0.0;
	if (y <= -1.36e-98)
		tmp = t_1;
	elseif (y <= 9e-149)
		tmp = (z / (t + x)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -1.36e-98], t$95$1, If[LessEqual[y, 9e-149], N[(N[(z / N[(t + x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.36000000000000003e-98 or 8.9999999999999996e-149 < y

   1. Initial program 53.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(a + z\right) - b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

      3. lower-+.f64 67.8

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]

   if -1.36000000000000003e-98 < y < 8.9999999999999996e-149

   1. Initial program 82.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 38.6

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

   5. Applied rewrites 38.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 37.9%

      \[\leadsto x \cdot \color{blue}{\frac{z}{x + t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.36 \cdot 10^{-98}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-149}:\\ \;\;\;\;\frac{z}{t + x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+29}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;z - \frac{b \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x 1e+29) (- (+ a z) b) (- z (/ (* b y) x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 1e+29) {
		tmp = (a + z) - b;
	} else {
		tmp = z - ((b * y) / x);
	}
	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 (x <= 1d+29) then
        tmp = (a + z) - b
    else
        tmp = z - ((b * y) / x)
    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 (x <= 1e+29) {
		tmp = (a + z) - b;
	} else {
		tmp = z - ((b * y) / x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= 1e+29:
		tmp = (a + z) - b
	else:
		tmp = z - ((b * y) / x)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= 1e+29)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = Float64(z - Float64(Float64(b * y) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= 1e+29)
		tmp = (a + z) - b;
	else
		tmp = z - ((b * y) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, 1e+29], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], N[(z - N[(N[(b * y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999914e28

   1. Initial program 61.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(a + z\right) - b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

      3. lower-+.f64 58.2

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   5. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]

   if 9.99999999999999914e28 < x

   1. Initial program 66.6%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 52.4%

      \[\leadsto z - \frac{b \cdot y}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+29}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;z - \frac{b \cdot y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+157}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;z - \frac{z}{x} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x 1.05e+157) (- (+ a z) b) (- z (* (/ z x) t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 1.05e+157) {
		tmp = (a + z) - b;
	} else {
		tmp = z - ((z / x) * t);
	}
	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 (x <= 1.05d+157) then
        tmp = (a + z) - b
    else
        tmp = z - ((z / x) * t)
    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 (x <= 1.05e+157) {
		tmp = (a + z) - b;
	} else {
		tmp = z - ((z / x) * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= 1.05e+157:
		tmp = (a + z) - b
	else:
		tmp = z - ((z / x) * t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= 1.05e+157)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = Float64(z - Float64(Float64(z / x) * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= 1.05e+157)
		tmp = (a + z) - b;
	else
		tmp = z - ((z / x) * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, 1.05e+157], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], N[(z - N[(N[(z / x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.05e157

   1. Initial program 64.2%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(a + z\right) - b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

      3. lower-+.f64 56.1

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   5. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]

   if 1.05e157 < x

   1. Initial program 50.1%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 61.3%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{+157}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;z - \frac{z}{x} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{-83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-88}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -5.5e-83) t_1 (if (<= y 2.35e-88) (+ a z) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double tmp;
	if (y <= -5.5e-83) {
		tmp = t_1;
	} else if (y <= 2.35e-88) {
		tmp = a + z;
	} 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 = (a + z) - b
    if (y <= (-5.5d-83)) then
        tmp = t_1
    else if (y <= 2.35d-88) then
        tmp = a + z
    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 = (a + z) - b;
	double tmp;
	if (y <= -5.5e-83) {
		tmp = t_1;
	} else if (y <= 2.35e-88) {
		tmp = a + z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a + z) - b
	tmp = 0
	if y <= -5.5e-83:
		tmp = t_1
	elif y <= 2.35e-88:
		tmp = a + z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -5.5e-83)
		tmp = t_1;
	elseif (y <= 2.35e-88)
		tmp = Float64(a + z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a + z) - b;
	tmp = 0.0;
	if (y <= -5.5e-83)
		tmp = t_1;
	elseif (y <= 2.35e-88)
		tmp = a + z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -5.5e-83], t$95$1, If[LessEqual[y, 2.35e-88], N[(a + z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.49999999999999964e-83 or 2.35e-88 < y

   1. Initial program 51.1%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(a + z\right) - b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

      3. lower-+.f64 69.2

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   5. Applied rewrites 69.2%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]

   if -5.49999999999999964e-83 < y < 2.35e-88

   1. Initial program 82.1%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(a + z\right) - b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

      3. lower-+.f64 28.9

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   5. Applied rewrites 28.9%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 40.0%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-83}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-88}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-51}:\\ \;\;\;\;z - b\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{-116}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2e-51) (- z b) (if (<= z 2.85e-116) (- a b) (+ a z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2e-51) {
		tmp = z - b;
	} else if (z <= 2.85e-116) {
		tmp = a - b;
	} else {
		tmp = a + z;
	}
	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 (z <= (-2d-51)) then
        tmp = z - b
    else if (z <= 2.85d-116) then
        tmp = a - b
    else
        tmp = a + z
    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 (z <= -2e-51) {
		tmp = z - b;
	} else if (z <= 2.85e-116) {
		tmp = a - b;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -2e-51:
		tmp = z - b
	elif z <= 2.85e-116:
		tmp = a - b
	else:
		tmp = a + z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2e-51)
		tmp = Float64(z - b);
	elseif (z <= 2.85e-116)
		tmp = Float64(a - b);
	else
		tmp = Float64(a + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -2e-51)
		tmp = z - b;
	elseif (z <= 2.85e-116)
		tmp = a - b;
	else
		tmp = a + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2e-51], N[(z - b), $MachinePrecision], If[LessEqual[z, 2.85e-116], N[(a - b), $MachinePrecision], N[(a + z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2e-51

   1. Initial program 65.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(a + z\right) - b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

      3. lower-+.f64 51.6

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   5. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 46.8%

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

   if -2e-51 < z < 2.8499999999999998e-116

   1. Initial program 66.8%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(a + z\right) - b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

      3. lower-+.f64 60.6

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   5. Applied rewrites 60.6%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 58.5%

      \[\leadsto a - \color{blue}{b} \]

   if 2.8499999999999998e-116 < z

   1. Initial program 54.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(a + z\right) - b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

      3. lower-+.f64 46.6

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   5. Applied rewrites 46.6%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 51.2%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-51}:\\ \;\;\;\;z - b\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{-116}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-52}:\\ \;\;\;\;a + z\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{-116}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.3e-52) (+ a z) (if (<= z 2.85e-116) (- a b) (+ a z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.3e-52) {
		tmp = a + z;
	} else if (z <= 2.85e-116) {
		tmp = a - b;
	} else {
		tmp = a + z;
	}
	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 (z <= (-2.3d-52)) then
        tmp = a + z
    else if (z <= 2.85d-116) then
        tmp = a - b
    else
        tmp = a + z
    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 (z <= -2.3e-52) {
		tmp = a + z;
	} else if (z <= 2.85e-116) {
		tmp = a - b;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -2.3e-52:
		tmp = a + z
	elif z <= 2.85e-116:
		tmp = a - b
	else:
		tmp = a + z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2.3e-52)
		tmp = Float64(a + z);
	elseif (z <= 2.85e-116)
		tmp = Float64(a - b);
	else
		tmp = Float64(a + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -2.3e-52)
		tmp = a + z;
	elseif (z <= 2.85e-116)
		tmp = a - b;
	else
		tmp = a + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2.3e-52], N[(a + z), $MachinePrecision], If[LessEqual[z, 2.85e-116], N[(a - b), $MachinePrecision], N[(a + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.29999999999999994e-52 or 2.8499999999999998e-116 < z

   1. Initial program 59.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(a + z\right) - b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

      3. lower-+.f64 49.0

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   5. Applied rewrites 49.0%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 47.3%

      \[\leadsto z + \color{blue}{a} \]

   if -2.29999999999999994e-52 < z < 2.8499999999999998e-116

   1. Initial program 66.8%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(a + z\right) - b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

      3. lower-+.f64 60.6

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   5. Applied rewrites 60.6%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 58.5%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-52}:\\ \;\;\;\;a + z\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{-116}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{+206}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;-b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= b 4.8e+206) (+ a z) (- b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= 4.8e+206) {
		tmp = a + z;
	} else {
		tmp = -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 (b <= 4.8d+206) then
        tmp = a + z
    else
        tmp = -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 (b <= 4.8e+206) {
		tmp = a + z;
	} else {
		tmp = -b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= 4.8e+206:
		tmp = a + z
	else:
		tmp = -b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= 4.8e+206)
		tmp = Float64(a + z);
	else
		tmp = Float64(-b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= 4.8e+206)
		tmp = a + z;
	else
		tmp = -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, 4.8e+206], N[(a + z), $MachinePrecision], (-b)]

Derivation

1. Split input into 2 regimes

2. if b < 4.7999999999999999e206

   1. Initial program 64.5%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(a + z\right) - b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

      3. lower-+.f64 54.3

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   5. Applied rewrites 54.3%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 48.8%

      \[\leadsto z + \color{blue}{a} \]

   if 4.7999999999999999e206 < b

   1. Initial program 41.2%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(a + z\right) - b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

      3. lower-+.f64 48.5

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   5. Applied rewrites 48.5%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]

   6. Taylor expanded in b around inf

      \[\leadsto -1 \cdot \color{blue}{b} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.5%

      \[\leadsto -b \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.8 \cdot 10^{+206}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;-b\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 13.4% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -b \end{array} \]

FPCore

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

C

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

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 = -b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := (-b)

Derivation

1. Initial program 62.8%

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

3. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   3. lower-+.f64 53.9

      \[\leadsto \color{blue}{\left(z + a\right)} - b \]

5. Applied rewrites 53.9%

   \[\leadsto \color{blue}{\left(z + a\right) - b} \]

6. Taylor expanded in b around inf

   \[\leadsto -1 \cdot \color{blue}{b} \]
7. Step-by-step derivation

8. Applied rewrites 16.0%

   \[\leadsto -b \]
9. Add Preprocessing

Developer Target 1: 82.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\\ t_3 := \frac{t\_2}{t\_1}\\ t_4 := \left(z + a\right) - b\\ \mathbf{if}\;t\_3 < -3.5813117084150564 \cdot 10^{+153}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 < 1.2285964308315609 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{t\_1}{t\_2}}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)))
        (t_3 (/ t_2 t_1))
        (t_4 (- (+ z a) b)))
   (if (< t_3 -3.5813117084150564e+153)
     t_4
     (if (< t_3 1.2285964308315609e+82) (/ 1.0 (/ t_1 t_2)) t_4))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b);
	double t_3 = t_2 / t_1;
	double t_4 = (z + a) - b;
	double tmp;
	if (t_3 < -3.5813117084150564e+153) {
		tmp = t_4;
	} else if (t_3 < 1.2285964308315609e+82) {
		tmp = 1.0 / (t_1 / t_2);
	} else {
		tmp = t_4;
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = (x + t) + y
    t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b)
    t_3 = t_2 / t_1
    t_4 = (z + a) - b
    if (t_3 < (-3.5813117084150564d+153)) then
        tmp = t_4
    else if (t_3 < 1.2285964308315609d+82) then
        tmp = 1.0d0 / (t_1 / t_2)
    else
        tmp = t_4
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b);
	double t_3 = t_2 / t_1;
	double t_4 = (z + a) - b;
	double tmp;
	if (t_3 < -3.5813117084150564e+153) {
		tmp = t_4;
	} else if (t_3 < 1.2285964308315609e+82) {
		tmp = 1.0 / (t_1 / t_2);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + t) + y
	t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b)
	t_3 = t_2 / t_1
	t_4 = (z + a) - b
	tmp = 0
	if t_3 < -3.5813117084150564e+153:
		tmp = t_4
	elif t_3 < 1.2285964308315609e+82:
		tmp = 1.0 / (t_1 / t_2)
	else:
		tmp = t_4
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b))
	t_3 = Float64(t_2 / t_1)
	t_4 = Float64(Float64(z + a) - b)
	tmp = 0.0
	if (t_3 < -3.5813117084150564e+153)
		tmp = t_4;
	elseif (t_3 < 1.2285964308315609e+82)
		tmp = Float64(1.0 / Float64(t_1 / t_2));
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x + t) + y;
	t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b);
	t_3 = t_2 / t_1;
	t_4 = (z + a) - b;
	tmp = 0.0;
	if (t_3 < -3.5813117084150564e+153)
		tmp = t_4;
	elseif (t_3 < 1.2285964308315609e+82)
		tmp = 1.0 / (t_1 / t_2);
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, If[Less[t$95$3, -3.5813117084150564e+153], t$95$4, If[Less[t$95$3, 1.2285964308315609e+82], N[(1.0 / N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]

Reproduce

herbie shell --seed 2024267 
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) -3581311708415056400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (+ z a) b) (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) 12285964308315609000000000000000000000000000000000000000000000000000000000000000000) (/ 1 (/ (+ (+ x t) y) (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)))) (- (+ z a) b))))

  (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))