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

Percentage Accurate: 61.1% → 93.6%

Time: 12.9s

Alternatives: 16

Speedup: 1.2×

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: 61.1% 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: 93.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \left(x + t\right)\\ t_2 := \frac{y}{t_1}\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{+124} \lor \neg \left(x \leq 1.4 \cdot 10^{+73}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t_1} + t_2, z, \frac{y}{\frac{t_1}{a - b}} + \frac{t \cdot a}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - b}{\frac{t_1}{y}} + \mathsf{fma}\left(a, t_2 + \frac{t}{t_1}, \frac{x \cdot z}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (+ x t))) (t_2 (/ y t_1)))
   (if (or (<= x -3.5e+124) (not (<= x 1.4e+73)))
     (fma (+ (/ x t_1) t_2) z (+ (/ y (/ t_1 (- a b))) (/ (* t a) t_1)))
     (+ (/ (- z b) (/ t_1 y)) (fma a (+ t_2 (/ t t_1)) (/ (* x z) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (x + t);
	double t_2 = y / t_1;
	double tmp;
	if ((x <= -3.5e+124) || !(x <= 1.4e+73)) {
		tmp = fma(((x / t_1) + t_2), z, ((y / (t_1 / (a - b))) + ((t * a) / t_1)));
	} else {
		tmp = ((z - b) / (t_1 / y)) + fma(a, (t_2 + (t / t_1)), ((x * z) / t_1));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(x + t))
	t_2 = Float64(y / t_1)
	tmp = 0.0
	if ((x <= -3.5e+124) || !(x <= 1.4e+73))
		tmp = fma(Float64(Float64(x / t_1) + t_2), z, Float64(Float64(y / Float64(t_1 / Float64(a - b))) + Float64(Float64(t * a) / t_1)));
	else
		tmp = Float64(Float64(Float64(z - b) / Float64(t_1 / y)) + fma(a, Float64(t_2 + Float64(t / t_1)), Float64(Float64(x * z) / t_1)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y / t$95$1), $MachinePrecision]}, If[Or[LessEqual[x, -3.5e+124], N[Not[LessEqual[x, 1.4e+73]], $MachinePrecision]], N[(N[(N[(x / t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] * z + N[(N[(y / N[(t$95$1 / N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t * a), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - b), $MachinePrecision] / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision] + N[(a * N[(t$95$2 + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(x * z), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5000000000000001e124 or 1.40000000000000004e73 < x

   1. Initial program 53.0%

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

      1. associate--l+ 53.0%

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

      2. fma-def 53.0%

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

      3. *-commutative 53.0%

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

      4. distribute-rgt-in 53.0%

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

      5. associate--l+ 53.0%

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

      6. fma-def 53.0%

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

      7. distribute-lft-out-- 53.3%

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

      8. +-commutative 53.3%

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

   3. Simplified 53.3%

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

   4. Taylor expanded in z around 0 70.2%

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

      1. fma-def 70.2%

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

      2. associate-/l* 95.3%

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

   6. Simplified 95.3%

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

   if -3.5000000000000001e124 < x < 1.40000000000000004e73

   1. Initial program 66.0%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in a around 0 81.0%

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

      1. associate-/l* 97.7%

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

      2. +-commutative 97.7%

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

      3. fma-def 97.7%

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

   6. Simplified 97.7%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+124} \lor \neg \left(x \leq 1.4 \cdot 10^{+73}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y + \left(x + t\right)} + \frac{y}{y + \left(x + t\right)}, z, \frac{y}{\frac{y + \left(x + t\right)}{a - b}} + \frac{t \cdot a}{y + \left(x + t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - b}{\frac{y + \left(x + t\right)}{y}} + \mathsf{fma}\left(a, \frac{y}{y + \left(x + t\right)} + \frac{t}{y + \left(x + t\right)}, \frac{x \cdot z}{y + \left(x + t\right)}\right)\\ \end{array} \]

Alternative 2: 91.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \left(x + t\right)\\ t_2 := \frac{z - b}{\frac{t_1}{y}}\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{+177}:\\ \;\;\;\;\frac{t \cdot a}{t_1} + \left(\frac{y}{\frac{t_1}{a - b}} + \frac{x + y}{\frac{t_1}{z}}\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+62}:\\ \;\;\;\;t_2 + \mathsf{fma}\left(a, \frac{y}{t_1} + \frac{t}{t_1}, \frac{x \cdot z}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(\frac{z}{\frac{x + t}{x}} + \frac{a}{\frac{x + t}{t}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (+ x t))) (t_2 (/ (- z b) (/ t_1 y))))
   (if (<= x -2.05e+177)
     (+ (/ (* t a) t_1) (+ (/ y (/ t_1 (- a b))) (/ (+ x y) (/ t_1 z))))
     (if (<= x 6.4e+62)
       (+ t_2 (fma a (+ (/ y t_1) (/ t t_1)) (/ (* x z) t_1)))
       (+ t_2 (+ (/ z (/ (+ x t) x)) (/ a (/ (+ x t) t))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (x + t);
	double t_2 = (z - b) / (t_1 / y);
	double tmp;
	if (x <= -2.05e+177) {
		tmp = ((t * a) / t_1) + ((y / (t_1 / (a - b))) + ((x + y) / (t_1 / z)));
	} else if (x <= 6.4e+62) {
		tmp = t_2 + fma(a, ((y / t_1) + (t / t_1)), ((x * z) / t_1));
	} else {
		tmp = t_2 + ((z / ((x + t) / x)) + (a / ((x + t) / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(x + t))
	t_2 = Float64(Float64(z - b) / Float64(t_1 / y))
	tmp = 0.0
	if (x <= -2.05e+177)
		tmp = Float64(Float64(Float64(t * a) / t_1) + Float64(Float64(y / Float64(t_1 / Float64(a - b))) + Float64(Float64(x + y) / Float64(t_1 / z))));
	elseif (x <= 6.4e+62)
		tmp = Float64(t_2 + fma(a, Float64(Float64(y / t_1) + Float64(t / t_1)), Float64(Float64(x * z) / t_1)));
	else
		tmp = Float64(t_2 + Float64(Float64(z / Float64(Float64(x + t) / x)) + Float64(a / Float64(Float64(x + t) / t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z - b), $MachinePrecision] / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.05e+177], N[(N[(N[(t * a), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(y / N[(t$95$1 / N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x + y), $MachinePrecision] / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.4e+62], N[(t$95$2 + N[(a * N[(N[(y / t$95$1), $MachinePrecision] + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(x * z), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(N[(z / N[(N[(x + t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(a / N[(N[(x + t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.05000000000000007e177

   1. Initial program 47.1%

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

      1. associate--l+ 47.1%

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

      2. fma-def 47.1%

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

      3. *-commutative 47.1%

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

      4. distribute-rgt-in 47.1%

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

      5. associate--l+ 47.1%

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

      6. fma-def 47.1%

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

      7. distribute-lft-out-- 47.8%

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

      8. +-commutative 47.8%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in z around inf 47.7%

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

      1. associate-+r+ 47.7%

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

      2. *-commutative 47.7%

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

      3. associate-/l* 66.6%

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

   6. Simplified 66.6%

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

   7. Taylor expanded in z around 0 66.6%

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

      1. associate-/l* 96.6%

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

      2. +-commutative 96.6%

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

      3. +-commutative 96.6%

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

   9. Simplified 96.6%

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

   if -2.05000000000000007e177 < x < 6.39999999999999968e62

   1. Initial program 67.5%

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

   3. Simplified 67.8%

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

   4. Taylor expanded in a around 0 81.2%

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

      1. associate-/l* 97.8%

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

      2. +-commutative 97.8%

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

      3. fma-def 97.8%

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

   6. Simplified 97.8%

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

   if 6.39999999999999968e62 < x

   1. Initial program 48.6%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in a around 0 61.5%

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

      1. associate-/l* 68.9%

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

      2. +-commutative 68.9%

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

      3. fma-def 68.9%

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

   6. Simplified 68.9%

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

   7. Taylor expanded in y around 0 54.8%

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

      1. +-commutative 54.8%

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

      2. associate-/l* 80.1%

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

      3. +-commutative 80.1%

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

      4. associate-/l* 90.8%

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

      5. +-commutative 90.8%

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

   9. Simplified 90.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{+177}:\\ \;\;\;\;\frac{t \cdot a}{y + \left(x + t\right)} + \left(\frac{y}{\frac{y + \left(x + t\right)}{a - b}} + \frac{x + y}{\frac{y + \left(x + t\right)}{z}}\right)\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{+62}:\\ \;\;\;\;\frac{z - b}{\frac{y + \left(x + t\right)}{y}} + \mathsf{fma}\left(a, \frac{y}{y + \left(x + t\right)} + \frac{t}{y + \left(x + t\right)}, \frac{x \cdot z}{y + \left(x + t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - b}{\frac{y + \left(x + t\right)}{y}} + \left(\frac{z}{\frac{x + t}{x}} + \frac{a}{\frac{x + t}{t}}\right)\\ \end{array} \]

Alternative 3: 92.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \left(x + t\right)\\ t_2 := a \cdot \left(y + t\right)\\ t_3 := \frac{\left(t_2 + z \cdot \left(x + y\right)\right) - y \cdot b}{t_1}\\ \mathbf{if}\;t_3 \leq -\infty \lor \neg \left(t_3 \leq 5 \cdot 10^{+253}\right):\\ \;\;\;\;\frac{z - b}{\frac{t_1}{y}} + \left(\frac{z}{\frac{x + t}{x}} + \frac{a}{\frac{x + t}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{x}{t_1} + \frac{y}{t_1}\right) \cdot z + \frac{t_2}{t_1}\right) - \frac{y \cdot b}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (+ x t)))
        (t_2 (* a (+ y t)))
        (t_3 (/ (- (+ t_2 (* z (+ x y))) (* y b)) t_1)))
   (if (or (<= t_3 (- INFINITY)) (not (<= t_3 5e+253)))
     (+ (/ (- z b) (/ t_1 y)) (+ (/ z (/ (+ x t) x)) (/ a (/ (+ x t) t))))
     (- (+ (* (+ (/ x t_1) (/ y t_1)) z) (/ t_2 t_1)) (/ (* y b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (x + t);
	double t_2 = a * (y + t);
	double t_3 = ((t_2 + (z * (x + y))) - (y * b)) / t_1;
	double tmp;
	if ((t_3 <= -((double) INFINITY)) || !(t_3 <= 5e+253)) {
		tmp = ((z - b) / (t_1 / y)) + ((z / ((x + t) / x)) + (a / ((x + t) / t)));
	} else {
		tmp = ((((x / t_1) + (y / t_1)) * z) + (t_2 / t_1)) - ((y * b) / t_1);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (x + t);
	double t_2 = a * (y + t);
	double t_3 = ((t_2 + (z * (x + y))) - (y * b)) / t_1;
	double tmp;
	if ((t_3 <= -Double.POSITIVE_INFINITY) || !(t_3 <= 5e+253)) {
		tmp = ((z - b) / (t_1 / y)) + ((z / ((x + t) / x)) + (a / ((x + t) / t)));
	} else {
		tmp = ((((x / t_1) + (y / t_1)) * z) + (t_2 / t_1)) - ((y * b) / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = y + (x + t)
	t_2 = a * (y + t)
	t_3 = ((t_2 + (z * (x + y))) - (y * b)) / t_1
	tmp = 0
	if (t_3 <= -math.inf) or not (t_3 <= 5e+253):
		tmp = ((z - b) / (t_1 / y)) + ((z / ((x + t) / x)) + (a / ((x + t) / t)))
	else:
		tmp = ((((x / t_1) + (y / t_1)) * z) + (t_2 / t_1)) - ((y * b) / t_1)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(x + t))
	t_2 = Float64(a * Float64(y + t))
	t_3 = Float64(Float64(Float64(t_2 + Float64(z * Float64(x + y))) - Float64(y * b)) / t_1)
	tmp = 0.0
	if ((t_3 <= Float64(-Inf)) || !(t_3 <= 5e+253))
		tmp = Float64(Float64(Float64(z - b) / Float64(t_1 / y)) + Float64(Float64(z / Float64(Float64(x + t) / x)) + Float64(a / Float64(Float64(x + t) / t))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(x / t_1) + Float64(y / t_1)) * z) + Float64(t_2 / t_1)) - Float64(Float64(y * b) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = y + (x + t);
	t_2 = a * (y + t);
	t_3 = ((t_2 + (z * (x + y))) - (y * b)) / t_1;
	tmp = 0.0;
	if ((t_3 <= -Inf) || ~((t_3 <= 5e+253)))
		tmp = ((z - b) / (t_1 / y)) + ((z / ((x + t) / x)) + (a / ((x + t) / t)));
	else
		tmp = ((((x / t_1) + (y / t_1)) * z) + (t_2 / t_1)) - ((y * b) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 + N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[Or[LessEqual[t$95$3, (-Infinity)], N[Not[LessEqual[t$95$3, 5e+253]], $MachinePrecision]], N[(N[(N[(z - b), $MachinePrecision] / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision] + N[(N[(z / N[(N[(x + t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(a / N[(N[(x + t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(x / t$95$1), $MachinePrecision] + N[(y / t$95$1), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] + N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(y * b), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

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 4.9999999999999997e253 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 12.2%

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

   3. Simplified 12.8%

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

   4. Taylor expanded in a around 0 39.6%

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

      1. associate-/l* 76.3%

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

      2. +-commutative 76.3%

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

      3. fma-def 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in y around 0 42.9%

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

      1. +-commutative 42.9%

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

      2. associate-/l* 62.2%

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

      3. +-commutative 62.2%

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

      4. associate-/l* 85.9%

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

      5. +-commutative 85.9%

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

   9. Simplified 85.9%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.9999999999999997e253

   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. Taylor expanded in z around 0 99.8%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(x + y\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty \lor \neg \left(\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(x + y\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq 5 \cdot 10^{+253}\right):\\ \;\;\;\;\frac{z - b}{\frac{y + \left(x + t\right)}{y}} + \left(\frac{z}{\frac{x + t}{x}} + \frac{a}{\frac{x + t}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{x}{y + \left(x + t\right)} + \frac{y}{y + \left(x + t\right)}\right) \cdot z + \frac{a \cdot \left(y + t\right)}{y + \left(x + t\right)}\right) - \frac{y \cdot b}{y + \left(x + t\right)}\\ \end{array} \]

Alternative 4: 92.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \left(x + t\right)\\ t_2 := \frac{\left(a \cdot \left(y + t\right) + z \cdot \left(x + y\right)\right) - y \cdot b}{t_1}\\ \mathbf{if}\;t_2 \leq -\infty \lor \neg \left(t_2 \leq 5 \cdot 10^{+253}\right):\\ \;\;\;\;\frac{z - b}{\frac{t_1}{y}} + \left(\frac{z}{\frac{x + t}{x}} + \frac{a}{\frac{x + t}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (+ x t)))
        (t_2 (/ (- (+ (* a (+ y t)) (* z (+ x y))) (* y b)) t_1)))
   (if (or (<= t_2 (- INFINITY)) (not (<= t_2 5e+253)))
     (+ (/ (- z b) (/ t_1 y)) (+ (/ z (/ (+ x t) x)) (/ a (/ (+ x t) t))))
     t_2)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (x + t);
	double t_2 = (((a * (y + t)) + (z * (x + y))) - (y * b)) / t_1;
	double tmp;
	if ((t_2 <= -((double) INFINITY)) || !(t_2 <= 5e+253)) {
		tmp = ((z - b) / (t_1 / y)) + ((z / ((x + t) / x)) + (a / ((x + t) / t)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (x + t);
	double t_2 = (((a * (y + t)) + (z * (x + y))) - (y * b)) / t_1;
	double tmp;
	if ((t_2 <= -Double.POSITIVE_INFINITY) || !(t_2 <= 5e+253)) {
		tmp = ((z - b) / (t_1 / y)) + ((z / ((x + t) / x)) + (a / ((x + t) / t)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = y + (x + t)
	t_2 = (((a * (y + t)) + (z * (x + y))) - (y * b)) / t_1
	tmp = 0
	if (t_2 <= -math.inf) or not (t_2 <= 5e+253):
		tmp = ((z - b) / (t_1 / y)) + ((z / ((x + t) / x)) + (a / ((x + t) / t)))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(x + t))
	t_2 = Float64(Float64(Float64(Float64(a * Float64(y + t)) + Float64(z * Float64(x + y))) - Float64(y * b)) / t_1)
	tmp = 0.0
	if ((t_2 <= Float64(-Inf)) || !(t_2 <= 5e+253))
		tmp = Float64(Float64(Float64(z - b) / Float64(t_1 / y)) + Float64(Float64(z / Float64(Float64(x + t) / x)) + Float64(a / Float64(Float64(x + t) / t))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = y + (x + t);
	t_2 = (((a * (y + t)) + (z * (x + y))) - (y * b)) / t_1;
	tmp = 0.0;
	if ((t_2 <= -Inf) || ~((t_2 <= 5e+253)))
		tmp = ((z - b) / (t_1 / y)) + ((z / ((x + t) / x)) + (a / ((x + t) / t)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision] + N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[Or[LessEqual[t$95$2, (-Infinity)], N[Not[LessEqual[t$95$2, 5e+253]], $MachinePrecision]], N[(N[(N[(z - b), $MachinePrecision] / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision] + N[(N[(z / N[(N[(x + t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(a / N[(N[(x + t), $MachinePrecision] / t), $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 4.9999999999999997e253 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 12.2%

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

   3. Simplified 12.8%

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

   4. Taylor expanded in a around 0 39.6%

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

      1. associate-/l* 76.3%

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

      2. +-commutative 76.3%

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

      3. fma-def 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in y around 0 42.9%

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

      1. +-commutative 42.9%

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

      2. associate-/l* 62.2%

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

      3. +-commutative 62.2%

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

      4. associate-/l* 85.9%

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

      5. +-commutative 85.9%

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

   9. Simplified 85.9%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.9999999999999997e253

   1. Initial program 99.7%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(x + y\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty \lor \neg \left(\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(x + y\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq 5 \cdot 10^{+253}\right):\\ \;\;\;\;\frac{z - b}{\frac{y + \left(x + t\right)}{y}} + \left(\frac{z}{\frac{x + t}{x}} + \frac{a}{\frac{x + t}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(x + y\right)\right) - y \cdot b}{y + \left(x + t\right)}\\ \end{array} \]

Alternative 5: 88.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \left(x + t\right)\\ t_2 := \frac{\left(a \cdot \left(y + t\right) + z \cdot \left(x + y\right)\right) - y \cdot b}{t_1}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;a + \frac{z - b}{\frac{t_1}{y}}\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (+ x t)))
        (t_2 (/ (- (+ (* a (+ y t)) (* z (+ x y))) (* y b)) t_1)))
   (if (<= t_2 (- INFINITY))
     (+ a (/ (- z b) (/ t_1 y)))
     (if (<= t_2 5e+286) t_2 (- (+ z a) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (x + t);
	double t_2 = (((a * (y + t)) + (z * (x + y))) - (y * b)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = a + ((z - b) / (t_1 / y));
	} else if (t_2 <= 5e+286) {
		tmp = t_2;
	} else {
		tmp = (z + a) - b;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (x + t);
	double t_2 = (((a * (y + t)) + (z * (x + y))) - (y * b)) / t_1;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = a + ((z - b) / (t_1 / y));
	} else if (t_2 <= 5e+286) {
		tmp = t_2;
	} else {
		tmp = (z + a) - b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = y + (x + t)
	t_2 = (((a * (y + t)) + (z * (x + y))) - (y * b)) / t_1
	tmp = 0
	if t_2 <= -math.inf:
		tmp = a + ((z - b) / (t_1 / y))
	elif t_2 <= 5e+286:
		tmp = t_2
	else:
		tmp = (z + a) - b
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(x + t))
	t_2 = Float64(Float64(Float64(Float64(a * Float64(y + t)) + Float64(z * Float64(x + y))) - Float64(y * b)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(a + Float64(Float64(z - b) / Float64(t_1 / y)));
	elseif (t_2 <= 5e+286)
		tmp = t_2;
	else
		tmp = Float64(Float64(z + a) - b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = y + (x + t);
	t_2 = (((a * (y + t)) + (z * (x + y))) - (y * b)) / t_1;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = a + ((z - b) / (t_1 / y));
	elseif (t_2 <= 5e+286)
		tmp = t_2;
	else
		tmp = (z + a) - b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision] + N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(a + N[(N[(z - b), $MachinePrecision] / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+286], t$95$2, N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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

   1. Initial program 6.7%

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

   3. Simplified 6.5%

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

   4. Taylor expanded in a around 0 38.3%

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

      1. associate-/l* 74.7%

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

      2. +-commutative 74.7%

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

      3. fma-def 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in y around inf 76.2%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5.0000000000000004e286

   1. Initial program 99.7%

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

   if 5.0000000000000004e286 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 7.6%

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

      1. associate--l+ 7.6%

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

      2. fma-def 8.2%

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

      3. *-commutative 8.2%

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

      4. distribute-rgt-in 8.0%

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

      5. associate--l+ 8.0%

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

      6. fma-def 8.1%

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

      7. distribute-lft-out-- 8.9%

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

      8. +-commutative 8.9%

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

   3. Simplified 8.9%

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

   4. Taylor expanded in y around inf 76.3%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(x + y\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty:\\ \;\;\;\;a + \frac{z - b}{\frac{y + \left(x + t\right)}{y}}\\ \mathbf{elif}\;\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(x + y\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(x + y\right)\right) - y \cdot b}{y + \left(x + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

Alternative 6: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \left(x + t\right)\\ t_2 := \frac{t \cdot a + x \cdot z}{x + t}\\ t_3 := a + \frac{z - b}{\frac{t_1}{y}}\\ \mathbf{if}\;y \leq -8.6 \cdot 10^{-44}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-161}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-96}:\\ \;\;\;\;\frac{z \cdot \left(x + y\right) - y \cdot b}{t_1}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-48}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (+ x t)))
        (t_2 (/ (+ (* t a) (* x z)) (+ x t)))
        (t_3 (+ a (/ (- z b) (/ t_1 y)))))
   (if (<= y -8.6e-44)
     t_3
     (if (<= y 1.35e-161)
       t_2
       (if (<= y 1.3e-96)
         (/ (- (* z (+ x y)) (* y b)) t_1)
         (if (<= y 1.4e-48) t_2 t_3))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (x + t);
	double t_2 = ((t * a) + (x * z)) / (x + t);
	double t_3 = a + ((z - b) / (t_1 / y));
	double tmp;
	if (y <= -8.6e-44) {
		tmp = t_3;
	} else if (y <= 1.35e-161) {
		tmp = t_2;
	} else if (y <= 1.3e-96) {
		tmp = ((z * (x + y)) - (y * b)) / t_1;
	} else if (y <= 1.4e-48) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	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) :: tmp
    t_1 = y + (x + t)
    t_2 = ((t * a) + (x * z)) / (x + t)
    t_3 = a + ((z - b) / (t_1 / y))
    if (y <= (-8.6d-44)) then
        tmp = t_3
    else if (y <= 1.35d-161) then
        tmp = t_2
    else if (y <= 1.3d-96) then
        tmp = ((z * (x + y)) - (y * b)) / t_1
    else if (y <= 1.4d-48) then
        tmp = t_2
    else
        tmp = t_3
    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 = y + (x + t);
	double t_2 = ((t * a) + (x * z)) / (x + t);
	double t_3 = a + ((z - b) / (t_1 / y));
	double tmp;
	if (y <= -8.6e-44) {
		tmp = t_3;
	} else if (y <= 1.35e-161) {
		tmp = t_2;
	} else if (y <= 1.3e-96) {
		tmp = ((z * (x + y)) - (y * b)) / t_1;
	} else if (y <= 1.4e-48) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = y + (x + t)
	t_2 = ((t * a) + (x * z)) / (x + t)
	t_3 = a + ((z - b) / (t_1 / y))
	tmp = 0
	if y <= -8.6e-44:
		tmp = t_3
	elif y <= 1.35e-161:
		tmp = t_2
	elif y <= 1.3e-96:
		tmp = ((z * (x + y)) - (y * b)) / t_1
	elif y <= 1.4e-48:
		tmp = t_2
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(x + t))
	t_2 = Float64(Float64(Float64(t * a) + Float64(x * z)) / Float64(x + t))
	t_3 = Float64(a + Float64(Float64(z - b) / Float64(t_1 / y)))
	tmp = 0.0
	if (y <= -8.6e-44)
		tmp = t_3;
	elseif (y <= 1.35e-161)
		tmp = t_2;
	elseif (y <= 1.3e-96)
		tmp = Float64(Float64(Float64(z * Float64(x + y)) - Float64(y * b)) / t_1);
	elseif (y <= 1.4e-48)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = y + (x + t);
	t_2 = ((t * a) + (x * z)) / (x + t);
	t_3 = a + ((z - b) / (t_1 / y));
	tmp = 0.0;
	if (y <= -8.6e-44)
		tmp = t_3;
	elseif (y <= 1.35e-161)
		tmp = t_2;
	elseif (y <= 1.3e-96)
		tmp = ((z * (x + y)) - (y * b)) / t_1;
	elseif (y <= 1.4e-48)
		tmp = t_2;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t * a), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(a + N[(N[(z - b), $MachinePrecision] / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.6e-44], t$95$3, If[LessEqual[y, 1.35e-161], t$95$2, If[LessEqual[y, 1.3e-96], N[(N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y, 1.4e-48], t$95$2, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.60000000000000027e-44 or 1.40000000000000002e-48 < y

   1. Initial program 46.5%

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

   3. Simplified 47.0%

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

   4. Taylor expanded in a around 0 63.1%

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

      1. associate-/l* 91.5%

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

      2. +-commutative 91.5%

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

      3. fma-def 91.5%

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

   6. Simplified 91.5%

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

   7. Taylor expanded in y around inf 85.3%

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

   if -8.60000000000000027e-44 < y < 1.35e-161 or 1.3000000000000001e-96 < y < 1.40000000000000002e-48

   1. Initial program 80.1%

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

      1. associate--l+ 80.1%

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

      2. fma-def 80.1%

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

      3. *-commutative 80.1%

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

      4. distribute-rgt-in 80.1%

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

      5. associate--l+ 80.1%

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

      6. fma-def 80.1%

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

      7. distribute-lft-out-- 80.1%

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

      8. +-commutative 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in y around 0 70.8%

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

   if 1.35e-161 < y < 1.3000000000000001e-96

   1. Initial program 91.9%

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

   2. Taylor expanded in a around 0 84.3%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{-44}:\\ \;\;\;\;a + \frac{z - b}{\frac{y + \left(x + t\right)}{y}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-161}:\\ \;\;\;\;\frac{t \cdot a + x \cdot z}{x + t}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-96}:\\ \;\;\;\;\frac{z \cdot \left(x + y\right) - y \cdot b}{y + \left(x + t\right)}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{t \cdot a + x \cdot z}{x + t}\\ \mathbf{else}:\\ \;\;\;\;a + \frac{z - b}{\frac{y + \left(x + t\right)}{y}}\\ \end{array} \]

Alternative 7: 62.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t \cdot a + x \cdot z}{x + t}\\ t_2 := \left(z + a\right) - b\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{-37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{-103}:\\ \;\;\;\;b \cdot \frac{-y}{x + \left(y + t\right)}\\ \mathbf{elif}\;y \leq 1.2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+27}:\\ \;\;\;\;\frac{y}{\frac{x + y}{a}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* t a) (* x z)) (+ x t))) (t_2 (- (+ z a) b)))
   (if (<= y -3.3e-37)
     t_2
     (if (<= y 5.3e-148)
       t_1
       (if (<= y 1.52e-103)
         (* b (/ (- y) (+ x (+ y t))))
         (if (<= y 1.2) t_1 (if (<= y 1.22e+27) (/ y (/ (+ x y) a)) t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((t * a) + (x * z)) / (x + t);
	double t_2 = (z + a) - b;
	double tmp;
	if (y <= -3.3e-37) {
		tmp = t_2;
	} else if (y <= 5.3e-148) {
		tmp = t_1;
	} else if (y <= 1.52e-103) {
		tmp = b * (-y / (x + (y + t)));
	} else if (y <= 1.2) {
		tmp = t_1;
	} else if (y <= 1.22e+27) {
		tmp = y / ((x + y) / a);
	} else {
		tmp = t_2;
	}
	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) :: tmp
    t_1 = ((t * a) + (x * z)) / (x + t)
    t_2 = (z + a) - b
    if (y <= (-3.3d-37)) then
        tmp = t_2
    else if (y <= 5.3d-148) then
        tmp = t_1
    else if (y <= 1.52d-103) then
        tmp = b * (-y / (x + (y + t)))
    else if (y <= 1.2d0) then
        tmp = t_1
    else if (y <= 1.22d+27) then
        tmp = y / ((x + y) / a)
    else
        tmp = t_2
    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 = ((t * a) + (x * z)) / (x + t);
	double t_2 = (z + a) - b;
	double tmp;
	if (y <= -3.3e-37) {
		tmp = t_2;
	} else if (y <= 5.3e-148) {
		tmp = t_1;
	} else if (y <= 1.52e-103) {
		tmp = b * (-y / (x + (y + t)));
	} else if (y <= 1.2) {
		tmp = t_1;
	} else if (y <= 1.22e+27) {
		tmp = y / ((x + y) / a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = ((t * a) + (x * z)) / (x + t)
	t_2 = (z + a) - b
	tmp = 0
	if y <= -3.3e-37:
		tmp = t_2
	elif y <= 5.3e-148:
		tmp = t_1
	elif y <= 1.52e-103:
		tmp = b * (-y / (x + (y + t)))
	elif y <= 1.2:
		tmp = t_1
	elif y <= 1.22e+27:
		tmp = y / ((x + y) / a)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(t * a) + Float64(x * z)) / Float64(x + t))
	t_2 = Float64(Float64(z + a) - b)
	tmp = 0.0
	if (y <= -3.3e-37)
		tmp = t_2;
	elseif (y <= 5.3e-148)
		tmp = t_1;
	elseif (y <= 1.52e-103)
		tmp = Float64(b * Float64(Float64(-y) / Float64(x + Float64(y + t))));
	elseif (y <= 1.2)
		tmp = t_1;
	elseif (y <= 1.22e+27)
		tmp = Float64(y / Float64(Float64(x + y) / a));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((t * a) + (x * z)) / (x + t);
	t_2 = (z + a) - b;
	tmp = 0.0;
	if (y <= -3.3e-37)
		tmp = t_2;
	elseif (y <= 5.3e-148)
		tmp = t_1;
	elseif (y <= 1.52e-103)
		tmp = b * (-y / (x + (y + t)));
	elseif (y <= 1.2)
		tmp = t_1;
	elseif (y <= 1.22e+27)
		tmp = y / ((x + y) / a);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(t * a), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -3.3e-37], t$95$2, If[LessEqual[y, 5.3e-148], t$95$1, If[LessEqual[y, 1.52e-103], N[(b * N[((-y) / N[(x + N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2], t$95$1, If[LessEqual[y, 1.22e+27], N[(y / N[(N[(x + y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.29999999999999982e-37 or 1.2200000000000001e27 < y

   1. Initial program 42.1%

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

      1. associate--l+ 42.1%

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

      2. fma-def 42.4%

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

      3. *-commutative 42.4%

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

      4. distribute-rgt-in 42.3%

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

      5. associate--l+ 42.3%

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

      6. fma-def 42.4%

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

      7. distribute-lft-out-- 42.7%

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

      8. +-commutative 42.7%

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

   3. Simplified 42.7%

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

   4. Taylor expanded in y around inf 72.7%

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

   if -3.29999999999999982e-37 < y < 5.29999999999999995e-148 or 1.52e-103 < y < 1.19999999999999996

   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. Step-by-step derivation

      1. associate--l+ 82.1%

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

      2. fma-def 82.1%

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

      3. *-commutative 82.1%

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

      4. distribute-rgt-in 82.1%

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

      5. associate--l+ 82.1%

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

      6. fma-def 82.1%

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

      7. distribute-lft-out-- 82.1%

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

      8. +-commutative 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in y around 0 69.1%

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

   if 5.29999999999999995e-148 < y < 1.52e-103

   1. Initial program 87.8%

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

      1. associate--l+ 87.8%

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

      2. fma-def 87.8%

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

      3. *-commutative 87.8%

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

      4. distribute-rgt-in 87.8%

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

      5. associate--l+ 87.8%

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

      6. fma-def 87.8%

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

      7. distribute-lft-out-- 87.8%

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

      8. +-commutative 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in b around inf 63.6%

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

      1. neg-mul-1 63.6%

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

      2. distribute-rgt-neg-in 63.6%

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

   6. Simplified 63.6%

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

      1. expm1-log1p-u 35.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y \cdot \left(-b\right)}{y + \left(x + t\right)}\right)\right)} \]

      2. expm1-udef 23.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y \cdot \left(-b\right)}{y + \left(x + t\right)}\right)} - 1} \]

      3. associate-/l* 3.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{y}{\frac{y + \left(x + t\right)}{-b}}}\right)} - 1 \]

      4. +-commutative 3.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{y}{\frac{\color{blue}{\left(x + t\right) + y}}{-b}}\right)} - 1 \]

   8. Applied egg-rr 3.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y}{\frac{\left(x + t\right) + y}{-b}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 15.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{\frac{\left(x + t\right) + y}{-b}}\right)\right)} \]

      2. expm1-log1p 40.6%

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

      3. associate-/r/ 63.6%

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

      4. associate-+l+ 63.6%

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

   10. Simplified 63.6%

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

   if 1.19999999999999996 < y < 1.2200000000000001e27

   1. Initial program 67.9%

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

      1. associate--l+ 67.9%

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

      2. fma-def 67.9%

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

      3. *-commutative 67.9%

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

      4. distribute-rgt-in 67.9%

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

      5. associate--l+ 67.9%

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

      6. fma-def 67.9%

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

      7. distribute-lft-out-- 67.9%

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

      8. +-commutative 67.9%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in a around inf 46.5%

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

   5. Taylor expanded in t around 0 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-/l* 78.3%

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

      3. +-commutative 78.3%

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

   7. Simplified 78.3%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{-37}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-148}:\\ \;\;\;\;\frac{t \cdot a + x \cdot z}{x + t}\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{-103}:\\ \;\;\;\;b \cdot \frac{-y}{x + \left(y + t\right)}\\ \mathbf{elif}\;y \leq 1.2:\\ \;\;\;\;\frac{t \cdot a + x \cdot z}{x + t}\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+27}:\\ \;\;\;\;\frac{y}{\frac{x + y}{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

Alternative 8: 71.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-41} \lor \neg \left(y \leq 1.9 \cdot 10^{-148}\right):\\ \;\;\;\;a + \frac{z - b}{\frac{y + \left(x + t\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot a + x \cdot z}{x + t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.1e-41) (not (<= y 1.9e-148)))
   (+ a (/ (- z b) (/ (+ y (+ x t)) y)))
   (/ (+ (* t a) (* x z)) (+ x t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.1e-41) || !(y <= 1.9e-148)) {
		tmp = a + ((z - b) / ((y + (x + t)) / y));
	} else {
		tmp = ((t * a) + (x * 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 ((y <= (-1.1d-41)) .or. (.not. (y <= 1.9d-148))) then
        tmp = a + ((z - b) / ((y + (x + t)) / y))
    else
        tmp = ((t * a) + (x * 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 ((y <= -1.1e-41) || !(y <= 1.9e-148)) {
		tmp = a + ((z - b) / ((y + (x + t)) / y));
	} else {
		tmp = ((t * a) + (x * z)) / (x + t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.1e-41) or not (y <= 1.9e-148):
		tmp = a + ((z - b) / ((y + (x + t)) / y))
	else:
		tmp = ((t * a) + (x * z)) / (x + t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.1e-41) || !(y <= 1.9e-148))
		tmp = Float64(a + Float64(Float64(z - b) / Float64(Float64(y + Float64(x + t)) / y)));
	else
		tmp = Float64(Float64(Float64(t * a) + Float64(x * z)) / Float64(x + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.1e-41) || ~((y <= 1.9e-148)))
		tmp = a + ((z - b) / ((y + (x + t)) / y));
	else
		tmp = ((t * a) + (x * z)) / (x + t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.1e-41], N[Not[LessEqual[y, 1.9e-148]], $MachinePrecision]], N[(a + N[(N[(z - b), $MachinePrecision] / N[(N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * a), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(x + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1e-41 or 1.90000000000000007e-148 < y

   1. Initial program 50.4%

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

   3. Simplified 50.8%

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

   4. Taylor expanded in a around 0 65.6%

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

      1. associate-/l* 90.7%

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

      2. +-commutative 90.7%

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

      3. fma-def 90.7%

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

   6. Simplified 90.7%

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

   7. Taylor expanded in y around inf 81.2%

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

   if -1.1e-41 < y < 1.90000000000000007e-148

   1. Initial program 81.7%

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

      1. associate--l+ 81.7%

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

      2. fma-def 81.7%

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

      3. *-commutative 81.7%

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

      4. distribute-rgt-in 81.7%

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

      5. associate--l+ 81.7%

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

      6. fma-def 81.7%

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

      7. distribute-lft-out-- 81.7%

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

      8. +-commutative 81.7%

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

   3. Simplified 81.7%

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

   4. Taylor expanded in y around 0 71.4%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-41} \lor \neg \left(y \leq 1.9 \cdot 10^{-148}\right):\\ \;\;\;\;a + \frac{z - b}{\frac{y + \left(x + t\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot a + x \cdot z}{x + t}\\ \end{array} \]

Alternative 9: 55.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - b}{\frac{y + t}{y}}\\ \mathbf{if}\;b \leq -3.15 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -5.3 \cdot 10^{+81}:\\ \;\;\;\;z\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+100}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- z b) (/ (+ y t) y))))
   (if (<= b -3.15e+153)
     t_1
     (if (<= b -5.3e+81) z (if (<= b 5.8e+100) (- (+ z a) b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z - b) / ((y + t) / y);
	double tmp;
	if (b <= -3.15e+153) {
		tmp = t_1;
	} else if (b <= -5.3e+81) {
		tmp = z;
	} else if (b <= 5.8e+100) {
		tmp = (z + a) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z - b) / ((y + t) / y)
    if (b <= (-3.15d+153)) then
        tmp = t_1
    else if (b <= (-5.3d+81)) then
        tmp = z
    else if (b <= 5.8d+100) then
        tmp = (z + a) - b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z - b) / ((y + t) / y);
	double tmp;
	if (b <= -3.15e+153) {
		tmp = t_1;
	} else if (b <= -5.3e+81) {
		tmp = z;
	} else if (b <= 5.8e+100) {
		tmp = (z + a) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (z - b) / ((y + t) / y)
	tmp = 0
	if b <= -3.15e+153:
		tmp = t_1
	elif b <= -5.3e+81:
		tmp = z
	elif b <= 5.8e+100:
		tmp = (z + a) - b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z - b) / Float64(Float64(y + t) / y))
	tmp = 0.0
	if (b <= -3.15e+153)
		tmp = t_1;
	elseif (b <= -5.3e+81)
		tmp = z;
	elseif (b <= 5.8e+100)
		tmp = Float64(Float64(z + a) - b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z - b) / ((y + t) / y);
	tmp = 0.0;
	if (b <= -3.15e+153)
		tmp = t_1;
	elseif (b <= -5.3e+81)
		tmp = z;
	elseif (b <= 5.8e+100)
		tmp = (z + a) - b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z - b), $MachinePrecision] / N[(N[(y + t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.15e+153], t$95$1, If[LessEqual[b, -5.3e+81], z, If[LessEqual[b, 5.8e+100], N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.1500000000000001e153 or 5.8000000000000001e100 < b

   1. Initial program 55.5%

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

   3. Simplified 56.0%

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

   4. Taylor expanded in a around 0 59.0%

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

      1. associate-/l* 87.5%

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

      2. +-commutative 87.5%

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

      3. fma-def 87.5%

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

   6. Simplified 87.5%

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

   7. Taylor expanded in a around 0 73.1%

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

      1. +-commutative 73.1%

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

   9. Simplified 73.1%

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

   10. Taylor expanded in x around 0 28.4%

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

       1. associate-/l* 52.1%

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

       2. +-commutative 52.1%

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

   12. Simplified 52.1%

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

   if -3.1500000000000001e153 < b < -5.30000000000000028e81

   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. Step-by-step derivation

      1. associate--l+ 64.5%

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

      2. fma-def 64.6%

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

      3. *-commutative 64.6%

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

      4. distribute-rgt-in 64.6%

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

      5. associate--l+ 64.6%

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

      6. fma-def 64.6%

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

      7. distribute-lft-out-- 64.7%

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

      8. +-commutative 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in x around inf 50.6%

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

   if -5.30000000000000028e81 < b < 5.8000000000000001e100

   1. Initial program 63.9%

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

      1. associate--l+ 63.9%

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

      2. fma-def 64.0%

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

      3. *-commutative 64.0%

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

      4. distribute-rgt-in 63.9%

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

      5. associate--l+ 63.9%

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

      6. fma-def 64.0%

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

      7. distribute-lft-out-- 64.1%

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

      8. +-commutative 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in y around inf 68.9%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.15 \cdot 10^{+153}:\\ \;\;\;\;\frac{z - b}{\frac{y + t}{y}}\\ \mathbf{elif}\;b \leq -5.3 \cdot 10^{+81}:\\ \;\;\;\;z\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{+100}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{z - b}{\frac{y + t}{y}}\\ \end{array} \]

Alternative 10: 58.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+152} \lor \neg \left(b \leq 2.4 \cdot 10^{+131}\right):\\ \;\;\;\;b \cdot \frac{-y}{x + \left(y + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -5.8e+152) (not (<= b 2.4e+131)))
   (* b (/ (- y) (+ x (+ y t))))
   (- (+ z a) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -5.8e+152) || !(b <= 2.4e+131)) {
		tmp = b * (-y / (x + (y + t)));
	} else {
		tmp = (z + a) - 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 <= (-5.8d+152)) .or. (.not. (b <= 2.4d+131))) then
        tmp = b * (-y / (x + (y + t)))
    else
        tmp = (z + a) - 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 <= -5.8e+152) || !(b <= 2.4e+131)) {
		tmp = b * (-y / (x + (y + t)));
	} else {
		tmp = (z + a) - b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -5.8e+152) or not (b <= 2.4e+131):
		tmp = b * (-y / (x + (y + t)))
	else:
		tmp = (z + a) - b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -5.8e+152) || !(b <= 2.4e+131))
		tmp = Float64(b * Float64(Float64(-y) / Float64(x + Float64(y + t))));
	else
		tmp = Float64(Float64(z + a) - b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -5.8e+152) || ~((b <= 2.4e+131)))
		tmp = b * (-y / (x + (y + t)));
	else
		tmp = (z + a) - b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -5.8e+152], N[Not[LessEqual[b, 2.4e+131]], $MachinePrecision]], N[(b * N[((-y) / N[(x + N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.7999999999999997e152 or 2.3999999999999999e131 < b

   1. Initial program 55.6%

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

      1. associate--l+ 55.6%

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

      2. fma-def 55.8%

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

      3. *-commutative 55.8%

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

      4. distribute-rgt-in 55.8%

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

      5. associate--l+ 55.8%

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

      6. fma-def 55.8%

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

      7. distribute-lft-out-- 56.1%

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

      8. +-commutative 56.1%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in b around inf 33.5%

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

      1. neg-mul-1 33.5%

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

      2. distribute-rgt-neg-in 33.5%

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

   6. Simplified 33.5%

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

      1. expm1-log1p-u 20.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y \cdot \left(-b\right)}{y + \left(x + t\right)}\right)\right)} \]

      2. expm1-udef 12.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y \cdot \left(-b\right)}{y + \left(x + t\right)}\right)} - 1} \]

      3. associate-/l* 22.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{y}{\frac{y + \left(x + t\right)}{-b}}}\right)} - 1 \]

      4. +-commutative 22.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{y}{\frac{\color{blue}{\left(x + t\right) + y}}{-b}}\right)} - 1 \]

   8. Applied egg-rr 22.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y}{\frac{\left(x + t\right) + y}{-b}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 29.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{\frac{\left(x + t\right) + y}{-b}}\right)\right)} \]

      2. expm1-log1p 59.7%

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

      3. associate-/r/ 62.3%

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

      4. associate-+l+ 62.3%

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

   10. Simplified 62.3%

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

   if -5.7999999999999997e152 < b < 2.3999999999999999e131

   1. Initial program 63.8%

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

      1. associate--l+ 63.8%

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

      2. fma-def 64.0%

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

      3. *-commutative 64.0%

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

      4. distribute-rgt-in 63.9%

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

      5. associate--l+ 63.9%

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

      6. fma-def 63.9%

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

      7. distribute-lft-out-- 64.1%

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

      8. +-commutative 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in y around inf 64.5%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+152} \lor \neg \left(b \leq 2.4 \cdot 10^{+131}\right):\\ \;\;\;\;b \cdot \frac{-y}{x + \left(y + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

Alternative 11: 54.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \frac{-b}{y + t}\\ \mathbf{elif}\;b \leq -5.3 \cdot 10^{+81}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -6.4e+153)
   (* y (/ (- b) (+ y t)))
   (if (<= b -5.3e+81) z (- (+ z a) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -6.4e+153) {
		tmp = y * (-b / (y + t));
	} else if (b <= -5.3e+81) {
		tmp = z;
	} else {
		tmp = (z + a) - 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 <= (-6.4d+153)) then
        tmp = y * (-b / (y + t))
    else if (b <= (-5.3d+81)) then
        tmp = z
    else
        tmp = (z + a) - 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 <= -6.4e+153) {
		tmp = y * (-b / (y + t));
	} else if (b <= -5.3e+81) {
		tmp = z;
	} else {
		tmp = (z + a) - b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -6.4e+153:
		tmp = y * (-b / (y + t))
	elif b <= -5.3e+81:
		tmp = z
	else:
		tmp = (z + a) - b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -6.4e+153)
		tmp = Float64(y * Float64(Float64(-b) / Float64(y + t)));
	elseif (b <= -5.3e+81)
		tmp = z;
	else
		tmp = Float64(Float64(z + a) - b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -6.4e+153)
		tmp = y * (-b / (y + t));
	elseif (b <= -5.3e+81)
		tmp = z;
	else
		tmp = (z + a) - b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -6.4e+153], N[(y * N[((-b) / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -5.3e+81], z, N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.4000000000000003e153

   1. Initial program 55.7%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in a around 0 59.6%

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

      1. associate-/l* 93.0%

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

      2. +-commutative 93.0%

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

      3. fma-def 93.0%

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

   6. Simplified 93.0%

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

   7. Taylor expanded in a around 0 75.9%

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

      1. +-commutative 75.9%

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

   9. Simplified 75.9%

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

   10. Taylor expanded in x around 0 29.7%

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

       1. associate-/l* 53.1%

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

       2. +-commutative 53.1%

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

   12. Simplified 53.1%

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

   13. Taylor expanded in z around 0 29.7%

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

       1. mul-1-neg 29.7%

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

       2. *-commutative 29.7%

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

       3. +-commutative 29.7%

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

       4. associate-*l/ 49.9%

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

       5. distribute-rgt-neg-in 49.9%

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

   15. Simplified 49.9%

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

   if -6.4000000000000003e153 < b < -5.30000000000000028e81

   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. Step-by-step derivation

      1. associate--l+ 64.5%

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

      2. fma-def 64.6%

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

      3. *-commutative 64.6%

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

      4. distribute-rgt-in 64.6%

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

      5. associate--l+ 64.6%

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

      6. fma-def 64.6%

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

      7. distribute-lft-out-- 64.7%

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

      8. +-commutative 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in x around inf 50.6%

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

   if -5.30000000000000028e81 < b

   1. Initial program 62.3%

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

      1. associate--l+ 62.3%

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

      2. fma-def 62.4%

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

      3. *-commutative 62.4%

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

      4. distribute-rgt-in 62.3%

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

      5. associate--l+ 62.3%

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

      6. fma-def 62.3%

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

      7. distribute-lft-out-- 62.5%

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

      8. +-commutative 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in y around inf 63.2%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{+153}:\\ \;\;\;\;y \cdot \frac{-b}{y + t}\\ \mathbf{elif}\;b \leq -5.3 \cdot 10^{+81}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

Alternative 12: 54.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{-y}{\frac{y + t}{b}}\\ \mathbf{elif}\;b \leq -5.3 \cdot 10^{+81}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -4.3e+154)
   (/ (- y) (/ (+ y t) b))
   (if (<= b -5.3e+81) z (- (+ z a) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -4.3e+154) {
		tmp = -y / ((y + t) / b);
	} else if (b <= -5.3e+81) {
		tmp = z;
	} else {
		tmp = (z + a) - 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 <= (-4.3d+154)) then
        tmp = -y / ((y + t) / b)
    else if (b <= (-5.3d+81)) then
        tmp = z
    else
        tmp = (z + a) - 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 <= -4.3e+154) {
		tmp = -y / ((y + t) / b);
	} else if (b <= -5.3e+81) {
		tmp = z;
	} else {
		tmp = (z + a) - b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -4.3e+154:
		tmp = -y / ((y + t) / b)
	elif b <= -5.3e+81:
		tmp = z
	else:
		tmp = (z + a) - b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -4.3e+154)
		tmp = Float64(Float64(-y) / Float64(Float64(y + t) / b));
	elseif (b <= -5.3e+81)
		tmp = z;
	else
		tmp = Float64(Float64(z + a) - b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -4.3e+154)
		tmp = -y / ((y + t) / b);
	elseif (b <= -5.3e+81)
		tmp = z;
	else
		tmp = (z + a) - b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -4.3e+154], N[((-y) / N[(N[(y + t), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -5.3e+81], z, N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.2999999999999998e154

   1. Initial program 55.7%

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

      1. associate--l+ 55.7%

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

      2. fma-def 56.3%

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

      3. *-commutative 56.3%

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

      4. distribute-rgt-in 56.3%

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

      5. associate--l+ 56.3%

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

      6. fma-def 56.3%

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

      7. distribute-lft-out-- 56.3%

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

      8. +-commutative 56.3%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in b around inf 39.1%

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

      1. neg-mul-1 39.1%

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

      2. distribute-rgt-neg-in 39.1%

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

   6. Simplified 39.1%

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

   7. Taylor expanded in x around 0 29.7%

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

      1. mul-1-neg 29.7%

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

      2. associate-/l* 50.0%

         \[\leadsto -\color{blue}{\frac{y}{\frac{y + t}{b}}} \]

      3. distribute-neg-frac 50.0%

         \[\leadsto \color{blue}{\frac{-y}{\frac{y + t}{b}}} \]

      4. +-commutative 50.0%

         \[\leadsto \frac{-y}{\frac{\color{blue}{t + y}}{b}} \]

   9. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{-y}{\frac{t + y}{b}}} \]

   if -4.2999999999999998e154 < b < -5.30000000000000028e81

   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. Step-by-step derivation

      1. associate--l+ 64.5%

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

      2. fma-def 64.6%

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

      3. *-commutative 64.6%

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

      4. distribute-rgt-in 64.6%

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

      5. associate--l+ 64.6%

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

      6. fma-def 64.6%

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

      7. distribute-lft-out-- 64.7%

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

      8. +-commutative 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in x around inf 50.6%

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

   if -5.30000000000000028e81 < b

   1. Initial program 62.3%

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

      1. associate--l+ 62.3%

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

      2. fma-def 62.4%

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

      3. *-commutative 62.4%

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

      4. distribute-rgt-in 62.3%

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

      5. associate--l+ 62.3%

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

      6. fma-def 62.3%

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

      7. distribute-lft-out-- 62.5%

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

      8. +-commutative 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in y around inf 63.2%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{-y}{\frac{y + t}{b}}\\ \mathbf{elif}\;b \leq -5.3 \cdot 10^{+81}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

Alternative 13: 48.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{+66}:\\ \;\;\;\;a\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-44}:\\ \;\;\;\;z - b\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -1.02e+66) a (if (<= a 6.2e-44) (- z b) a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.02e+66) {
		tmp = a;
	} else if (a <= 6.2e-44) {
		tmp = z - b;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.02d+66)) then
        tmp = a
    else if (a <= 6.2d-44) then
        tmp = z - b
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -1.02e+66) {
		tmp = a;
	} else if (a <= 6.2e-44) {
		tmp = z - b;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -1.02e+66:
		tmp = a
	elif a <= 6.2e-44:
		tmp = z - b
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -1.02e+66)
		tmp = a;
	elseif (a <= 6.2e-44)
		tmp = Float64(z - b);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -1.02e+66)
		tmp = a;
	elseif (a <= 6.2e-44)
		tmp = z - b;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -1.02e+66], a, If[LessEqual[a, 6.2e-44], N[(z - b), $MachinePrecision], a]]

Derivation

1. Split input into 2 regimes

2. if a < -1.01999999999999998e66 or 6.19999999999999968e-44 < a

   1. Initial program 49.9%

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

      1. associate--l+ 49.9%

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

      2. fma-def 50.1%

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

      3. *-commutative 50.1%

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

      4. distribute-rgt-in 50.0%

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

      5. associate--l+ 50.0%

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

      6. fma-def 50.0%

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

      7. distribute-lft-out-- 50.4%

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

      8. +-commutative 50.4%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in t around inf 50.2%

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

   if -1.01999999999999998e66 < a < 6.19999999999999968e-44

   1. Initial program 72.8%

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

   3. Simplified 73.0%

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

   4. Taylor expanded in a around 0 73.7%

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

      1. associate-/l* 90.9%

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

      2. +-commutative 90.9%

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

      3. fma-def 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in a around 0 76.0%

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

      1. +-commutative 76.0%

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

   9. Simplified 76.0%

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

   10. Taylor expanded in y around inf 49.6%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{+66}:\\ \;\;\;\;a\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-44}:\\ \;\;\;\;z - b\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 14: 58.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+148}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= x 4.6e+148) (- (+ z a) b) z))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= 4.6e+148) {
		tmp = (z + a) - b;
	} else {
		tmp = 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 (x <= 4.6d+148) then
        tmp = (z + a) - b
    else
        tmp = 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 (x <= 4.6e+148) {
		tmp = (z + a) - b;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= 4.6e+148:
		tmp = (z + a) - b
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= 4.6e+148)
		tmp = Float64(Float64(z + a) - b);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= 4.6e+148)
		tmp = (z + a) - b;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, 4.6e+148], N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision], z]

Derivation

1. Split input into 2 regimes

2. if x < 4.6000000000000001e148

   1. Initial program 64.0%

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

      1. associate--l+ 64.0%

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

      2. fma-def 64.1%

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

      3. *-commutative 64.1%

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

      4. distribute-rgt-in 64.1%

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

      5. associate--l+ 64.1%

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

      6. fma-def 64.1%

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

      7. distribute-lft-out-- 64.4%

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

      8. +-commutative 64.4%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in y around inf 59.5%

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

   if 4.6000000000000001e148 < x

   1. Initial program 42.2%

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

      1. associate--l+ 42.2%

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

      2. fma-def 42.2%

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

      3. *-commutative 42.2%

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

      4. distribute-rgt-in 42.0%

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

      5. associate--l+ 42.0%

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

      6. fma-def 42.0%

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

      7. distribute-lft-out-- 42.0%

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

      8. +-commutative 42.0%

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

   3. Simplified 42.0%

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

   4. Taylor expanded in x around inf 67.9%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+148}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 15: 43.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-71}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+62}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.5e-71) z (if (<= z 3.2e+62) a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.5e-71) {
		tmp = z;
	} else if (z <= 3.2e+62) {
		tmp = a;
	} else {
		tmp = 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 <= (-2.5d-71)) then
        tmp = z
    else if (z <= 3.2d+62) then
        tmp = a
    else
        tmp = 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 <= -2.5e-71) {
		tmp = z;
	} else if (z <= 3.2e+62) {
		tmp = a;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -2.5e-71:
		tmp = z
	elif z <= 3.2e+62:
		tmp = a
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2.5e-71)
		tmp = z;
	elseif (z <= 3.2e+62)
		tmp = a;
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -2.5e-71)
		tmp = z;
	elseif (z <= 3.2e+62)
		tmp = a;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2.5e-71], z, If[LessEqual[z, 3.2e+62], a, z]]

Derivation

1. Split input into 2 regimes

2. if z < -2.49999999999999999e-71 or 3.19999999999999984e62 < z

   1. Initial program 50.7%

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

      1. associate--l+ 50.7%

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

      2. fma-def 51.0%

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

      3. *-commutative 51.0%

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

      4. distribute-rgt-in 50.9%

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

      5. associate--l+ 50.9%

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

      6. fma-def 51.0%

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

      7. distribute-lft-out-- 51.2%

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

      8. +-commutative 51.2%

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

   3. Simplified 51.2%

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

   4. Taylor expanded in x around inf 54.7%

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

   if -2.49999999999999999e-71 < z < 3.19999999999999984e62

   1. Initial program 72.2%

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

      1. associate--l+ 72.2%

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

      2. fma-def 72.2%

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

      3. *-commutative 72.2%

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

      4. distribute-rgt-in 72.2%

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

      5. associate--l+ 72.2%

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

      6. fma-def 72.2%

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

      7. distribute-lft-out-- 72.3%

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

      8. +-commutative 72.3%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in t around inf 42.9%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-71}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+62}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 16: 32.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := a

Derivation

1. Initial program 61.8%

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

   1. associate--l+ 61.8%

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

   2. fma-def 61.9%

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

   3. *-commutative 61.9%

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

   4. distribute-rgt-in 61.9%

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

   5. associate--l+ 61.9%

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

   6. fma-def 61.9%

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

   7. distribute-lft-out-- 62.1%

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

   8. +-commutative 62.1%

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

3. Simplified 62.1%

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

4. Taylor expanded in t around inf 32.1%

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

5. Final simplification 32.1%

   \[\leadsto a \]

Developer target: 83.1% 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 2023207 
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"
  :precision binary64

  :herbie-target
  (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) -3.5813117084150564e+153) (- (+ z a) b) (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) 1.2285964308315609e+82) (/ 1.0 (/ (+ (+ 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)))