Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.3% → 89.4%

Time: 14.4s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y} - \frac{\mathsf{fma}\left(-y, \frac{x}{b - y}, \frac{y}{{\left(b - y\right)}^{2}} \cdot \left(t - a\right)\right)}{z}\\ \mathbf{if}\;z \leq -8.8 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2500000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (-
          (/ (- t a) (- b y))
          (/
           (fma (- y) (/ x (- b y)) (* (/ y (pow (- b y) 2.0)) (- t a)))
           z))))
   (if (<= z -8.8e+15)
     t_1
     (if (<= z 2500000000.0)
       (/ (fma y x (* (- t a) z)) (+ (* (- b y) z) y))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((t - a) / (b - y)) - (fma(-y, (x / (b - y)), ((y / pow((b - y), 2.0)) * (t - a))) / z);
	double tmp;
	if (z <= -8.8e+15) {
		tmp = t_1;
	} else if (z <= 2500000000.0) {
		tmp = fma(y, x, ((t - a) * z)) / (((b - y) * z) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(t - a) / Float64(b - y)) - Float64(fma(Float64(-y), Float64(x / Float64(b - y)), Float64(Float64(y / (Float64(b - y) ^ 2.0)) * Float64(t - a))) / z))
	tmp = 0.0
	if (z <= -8.8e+15)
		tmp = t_1;
	elseif (z <= 2500000000.0)
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / Float64(Float64(Float64(b - y) * z) + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(N[((-y) * N[(x / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(N[(y / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.8e+15], t$95$1, If[LessEqual[z, 2500000000.0], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.8e15 or 2.5e9 < z

   1. Initial program 40.6%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 94.3%

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

   if -8.8e15 < z < 2.5e9

   1. Initial program 93.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 93.3

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 93.3

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

   4. Applied rewrites 93.3%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{t - a}{b - y} - \frac{\mathsf{fma}\left(-y, \frac{x}{b - y}, \frac{y}{{\left(b - y\right)}^{2}} \cdot \left(t - a\right)\right)}{z}\\ \mathbf{elif}\;z \leq 2500000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y} - \frac{\mathsf{fma}\left(-y, \frac{x}{b - y}, \frac{y}{{\left(b - y\right)}^{2}} \cdot \left(t - a\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -62000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-207}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{t\_1}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-177}:\\ \;\;\;\;\frac{y}{t\_1} \cdot x\\ \mathbf{elif}\;z \leq 15000000000:\\ \;\;\;\;\frac{z}{t\_1} \cdot \left(t - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -62000000000.0)
     t_2
     (if (<= z -1.5e-207)
       (/ (* (- t a) z) t_1)
       (if (<= z 3.2e-177)
         (* (/ y t_1) x)
         (if (<= z 15000000000.0) (* (/ z t_1) (- t a)) t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -62000000000.0) {
		tmp = t_2;
	} else if (z <= -1.5e-207) {
		tmp = ((t - a) * z) / t_1;
	} else if (z <= 3.2e-177) {
		tmp = (y / t_1) * x;
	} else if (z <= 15000000000.0) {
		tmp = (z / t_1) * (t - a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -62000000000.0)
		tmp = t_2;
	elseif (z <= -1.5e-207)
		tmp = Float64(Float64(Float64(t - a) * z) / t_1);
	elseif (z <= 3.2e-177)
		tmp = Float64(Float64(y / t_1) * x);
	elseif (z <= 15000000000.0)
		tmp = Float64(Float64(z / t_1) * Float64(t - a));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -62000000000.0], t$95$2, If[LessEqual[z, -1.5e-207], N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 3.2e-177], N[(N[(y / t$95$1), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 15000000000.0], N[(N[(z / t$95$1), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.2e10 or 1.5e10 < z

   1. Initial program 40.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 83.9

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

   5. Applied rewrites 83.9%

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

   if -6.2e10 < z < -1.5e-207

   1. Initial program 92.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 92.3

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 92.3

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 68.3

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

   7. Applied rewrites 68.3%

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

   if -1.5e-207 < z < 3.1999999999999998e-177

   1. Initial program 94.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 73.5

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

   5. Applied rewrites 73.5%

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

   if 3.1999999999999998e-177 < z < 1.5e10

   1. Initial program 93.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 76.5

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

   5. Applied rewrites 76.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -62000000000:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -1.5 \cdot 10^{-207}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-177}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{elif}\;z \leq 15000000000:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \left(t - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4 \cdot 10^{+22}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-168}:\\ \;\;\;\;\frac{t - a}{t\_1} \cdot z\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-177}:\\ \;\;\;\;\frac{y}{t\_1} \cdot x\\ \mathbf{elif}\;z \leq 15000000000:\\ \;\;\;\;\frac{z}{t\_1} \cdot \left(t - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -4e+22)
     t_2
     (if (<= z -1.7e-168)
       (* (/ (- t a) t_1) z)
       (if (<= z 3.2e-177)
         (* (/ y t_1) x)
         (if (<= z 15000000000.0) (* (/ z t_1) (- t a)) t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -4e+22) {
		tmp = t_2;
	} else if (z <= -1.7e-168) {
		tmp = ((t - a) / t_1) * z;
	} else if (z <= 3.2e-177) {
		tmp = (y / t_1) * x;
	} else if (z <= 15000000000.0) {
		tmp = (z / t_1) * (t - a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -4e+22)
		tmp = t_2;
	elseif (z <= -1.7e-168)
		tmp = Float64(Float64(Float64(t - a) / t_1) * z);
	elseif (z <= 3.2e-177)
		tmp = Float64(Float64(y / t_1) * x);
	elseif (z <= 15000000000.0)
		tmp = Float64(Float64(z / t_1) * Float64(t - a));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4e+22], t$95$2, If[LessEqual[z, -1.7e-168], N[(N[(N[(t - a), $MachinePrecision] / t$95$1), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 3.2e-177], N[(N[(y / t$95$1), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 15000000000.0], N[(N[(z / t$95$1), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4e22 or 1.5e10 < z

   1. Initial program 40.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 83.7

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

   5. Applied rewrites 83.7%

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

   if -4e22 < z < -1.70000000000000011e-168

   1. Initial program 91.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 68.0

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

   5. Applied rewrites 68.0%

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

   7. Applied rewrites 70.3%

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

   if -1.70000000000000011e-168 < z < 3.1999999999999998e-177

   1. Initial program 93.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 70.7

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

   5. Applied rewrites 70.7%

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

   if 3.1999999999999998e-177 < z < 1.5e10

   1. Initial program 93.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 76.5

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

   5. Applied rewrites 76.5%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-168}:\\ \;\;\;\;\frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)} \cdot z\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-177}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{elif}\;z \leq 15000000000:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \left(t - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{t - a}{t\_1} \cdot z\\ t_3 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4 \cdot 10^{+22}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-168}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{y}{t\_1} \cdot x\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+19}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y))
        (t_2 (* (/ (- t a) t_1) z))
        (t_3 (/ (- t a) (- b y))))
   (if (<= z -4e+22)
     t_3
     (if (<= z -1.7e-168)
       t_2
       (if (<= z 7.5e-177) (* (/ y t_1) x) (if (<= z 2.4e+19) t_2 t_3))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = ((t - a) / t_1) * z;
	double t_3 = (t - a) / (b - y);
	double tmp;
	if (z <= -4e+22) {
		tmp = t_3;
	} else if (z <= -1.7e-168) {
		tmp = t_2;
	} else if (z <= 7.5e-177) {
		tmp = (y / t_1) * x;
	} else if (z <= 2.4e+19) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(Float64(t - a) / t_1) * z)
	t_3 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -4e+22)
		tmp = t_3;
	elseif (z <= -1.7e-168)
		tmp = t_2;
	elseif (z <= 7.5e-177)
		tmp = Float64(Float64(y / t_1) * x);
	elseif (z <= 2.4e+19)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t - a), $MachinePrecision] / t$95$1), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4e+22], t$95$3, If[LessEqual[z, -1.7e-168], t$95$2, If[LessEqual[z, 7.5e-177], N[(N[(y / t$95$1), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 2.4e+19], t$95$2, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4e22 or 2.4e19 < z

   1. Initial program 40.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 83.9

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

   5. Applied rewrites 83.9%

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

   if -4e22 < z < -1.70000000000000011e-168 or 7.5e-177 < z < 2.4e19

   1. Initial program 90.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 71.4

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

   5. Applied rewrites 71.4%

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

   7. Applied rewrites 71.8%

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

   if -1.70000000000000011e-168 < z < 7.5e-177

   1. Initial program 93.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 70.7

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

   5. Applied rewrites 70.7%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-168}:\\ \;\;\;\;\frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)} \cdot z\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+19}:\\ \;\;\;\;\frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)} \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -6.6 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+71}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -6.6e+16)
     t_1
     (if (<= z 6.2e+71) (/ (fma y x (* (- t a) z)) (+ (* (- b y) z) y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -6.6e+16) {
		tmp = t_1;
	} else if (z <= 6.2e+71) {
		tmp = fma(y, x, ((t - a) * z)) / (((b - y) * z) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -6.6e+16)
		tmp = t_1;
	elseif (z <= 6.2e+71)
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / Float64(Float64(Float64(b - y) * z) + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.6e+16], t$95$1, If[LessEqual[z, 6.2e+71], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.6e16 or 6.20000000000000036e71 < z

   1. Initial program 35.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if -6.6e16 < z < 6.20000000000000036e71

   1. Initial program 92.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 92.5

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 92.5

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

   4. Applied rewrites 92.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+16}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+71}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -36:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{-173}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{t\_1}\\ \mathbf{elif}\;z \leq 15000000000:\\ \;\;\;\;\frac{z}{t\_1} \cdot \left(t - a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -36.0)
     t_2
     (if (<= z 1.18e-173)
       (/ (fma t z (* y x)) t_1)
       (if (<= z 15000000000.0) (* (/ z t_1) (- t a)) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -36.0) {
		tmp = t_2;
	} else if (z <= 1.18e-173) {
		tmp = fma(t, z, (y * x)) / t_1;
	} else if (z <= 15000000000.0) {
		tmp = (z / t_1) * (t - a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -36.0)
		tmp = t_2;
	elseif (z <= 1.18e-173)
		tmp = Float64(fma(t, z, Float64(y * x)) / t_1);
	elseif (z <= 15000000000.0)
		tmp = Float64(Float64(z / t_1) * Float64(t - a));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -36.0], t$95$2, If[LessEqual[z, 1.18e-173], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 15000000000.0], N[(N[(z / t$95$1), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -36 or 1.5e10 < z

   1. Initial program 41.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 84.2

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

   5. Applied rewrites 84.2%

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

   if -36 < z < 1.1800000000000001e-173

   1. Initial program 93.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if 1.1800000000000001e-173 < z < 1.5e10

   1. Initial program 93.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 76.5

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

   5. Applied rewrites 76.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -36:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{-173}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{elif}\;z \leq 15000000000:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \left(t - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3700000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{b \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3700000000.0)
     t_1
     (if (<= z 9000.0) (/ (fma y x (* (- t a) z)) (+ (* b z) y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3700000000.0) {
		tmp = t_1;
	} else if (z <= 9000.0) {
		tmp = fma(y, x, ((t - a) * z)) / ((b * z) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -3700000000.0)
		tmp = t_1;
	elseif (z <= 9000.0)
		tmp = Float64(fma(y, x, Float64(Float64(t - a) * z)) / Float64(Float64(b * z) + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3700000000.0], t$95$1, If[LessEqual[z, 9000.0], N[(N[(y * x + N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(N[(b * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7e9 or 9e3 < z

   1. Initial program 40.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 83.9

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

   5. Applied rewrites 83.9%

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

   if -3.7e9 < z < 9e3

   1. Initial program 93.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 93.3

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 92.1

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

   7. Applied rewrites 92.1%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3700000000:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 9000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, \left(t - a\right) \cdot z\right)}{b \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4.9 \cdot 10^{-63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-135}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4.9e-63)
     t_1
     (if (<= z 4.4e-135) (* (/ y (fma (- b y) z y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -4.9e-63) {
		tmp = t_1;
	} else if (z <= 4.4e-135) {
		tmp = (y / fma((b - y), z, y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -4.9e-63)
		tmp = t_1;
	elseif (z <= 4.4e-135)
		tmp = Float64(Float64(y / fma(Float64(b - y), z, y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.9e-63], t$95$1, If[LessEqual[z, 4.4e-135], N[(N[(y / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.90000000000000015e-63 or 4.3999999999999999e-135 < z

   1. Initial program 54.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.6

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

   5. Applied rewrites 76.6%

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

   if -4.90000000000000015e-63 < z < 4.3999999999999999e-135

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 59.0

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

   5. Applied rewrites 59.0%

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

Alternative 9: 61.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -5.7 \cdot 10^{-75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.65 \cdot 10^{-207}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{y}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-135}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -5.7e-75)
     t_1
     (if (<= z -2.65e-207)
       (/ (* (- t a) z) y)
       (if (<= z 1.8e-135) (* 1.0 x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -5.7e-75) {
		tmp = t_1;
	} else if (z <= -2.65e-207) {
		tmp = ((t - a) * z) / y;
	} else if (z <= 1.8e-135) {
		tmp = 1.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-5.7d-75)) then
        tmp = t_1
    else if (z <= (-2.65d-207)) then
        tmp = ((t - a) * z) / y
    else if (z <= 1.8d-135) then
        tmp = 1.0d0 * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -5.7e-75) {
		tmp = t_1;
	} else if (z <= -2.65e-207) {
		tmp = ((t - a) * z) / y;
	} else if (z <= 1.8e-135) {
		tmp = 1.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -5.7e-75:
		tmp = t_1
	elif z <= -2.65e-207:
		tmp = ((t - a) * z) / y
	elif z <= 1.8e-135:
		tmp = 1.0 * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -5.7e-75)
		tmp = t_1;
	elseif (z <= -2.65e-207)
		tmp = Float64(Float64(Float64(t - a) * z) / y);
	elseif (z <= 1.8e-135)
		tmp = Float64(1.0 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -5.7e-75)
		tmp = t_1;
	elseif (z <= -2.65e-207)
		tmp = ((t - a) * z) / y;
	elseif (z <= 1.8e-135)
		tmp = 1.0 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.7e-75], t$95$1, If[LessEqual[z, -2.65e-207], N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, 1.8e-135], N[(1.0 * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.69999999999999966e-75 or 1.79999999999999989e-135 < z

   1. Initial program 55.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 75.6

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

   5. Applied rewrites 75.6%

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

   if -5.69999999999999966e-75 < z < -2.65e-207

   1. Initial program 91.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 61.4

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

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 48.9%

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

   if -2.65e-207 < z < 1.79999999999999989e-135

   1. Initial program 93.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 93.3

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 68.3

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

   7. Applied rewrites 68.3%

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

   8. Taylor expanded in z around 0

      \[\leadsto x \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 58.7%

       \[\leadsto x \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.7 \cdot 10^{-75}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -2.65 \cdot 10^{-207}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{y}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-135}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{-63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-135}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4.8e-63) t_1 (if (<= z 1.8e-135) (* 1.0 x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -4.8e-63) {
		tmp = t_1;
	} else if (z <= 1.8e-135) {
		tmp = 1.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-4.8d-63)) then
        tmp = t_1
    else if (z <= 1.8d-135) then
        tmp = 1.0d0 * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -4.8e-63) {
		tmp = t_1;
	} else if (z <= 1.8e-135) {
		tmp = 1.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -4.8e-63:
		tmp = t_1
	elif z <= 1.8e-135:
		tmp = 1.0 * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -4.8e-63)
		tmp = t_1;
	elseif (z <= 1.8e-135)
		tmp = Float64(1.0 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -4.8e-63)
		tmp = t_1;
	elseif (z <= 1.8e-135)
		tmp = 1.0 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.8e-63], t$95$1, If[LessEqual[z, 1.8e-135], N[(1.0 * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8000000000000001e-63 or 1.79999999999999989e-135 < z

   1. Initial program 54.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.6

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

   5. Applied rewrites 76.6%

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

   if -4.8000000000000001e-63 < z < 1.79999999999999989e-135

   1. Initial program 92.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 92.1

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 92.1

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 59.0

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

   7. Applied rewrites 59.0%

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

   8. Taylor expanded in z around 0

      \[\leadsto x \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 50.3%

       \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-135}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -1.3e+52) t_1 (if (<= y 6.5e+103) (/ (- t a) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -1.3e+52) {
		tmp = t_1;
	} else if (y <= 6.5e+103) {
		tmp = (t - 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 = x / (1.0d0 - z)
    if (y <= (-1.3d+52)) then
        tmp = t_1
    else if (y <= 6.5d+103) then
        tmp = (t - 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 = x / (1.0 - z);
	double tmp;
	if (y <= -1.3e+52) {
		tmp = t_1;
	} else if (y <= 6.5e+103) {
		tmp = (t - a) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -1.3e+52:
		tmp = t_1
	elif y <= 6.5e+103:
		tmp = (t - a) / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -1.3e+52)
		tmp = t_1;
	elseif (y <= 6.5e+103)
		tmp = Float64(Float64(t - a) / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -1.3e+52)
		tmp = t_1;
	elseif (y <= 6.5e+103)
		tmp = (t - a) / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.3e+52], t$95$1, If[LessEqual[y, 6.5e+103], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e52 or 6.50000000000000001e103 < y

   1. Initial program 52.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{x}{1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}} \]

      3. unsub-neg N/A

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

      4. lower--.f64 54.7

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

   5. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\frac{x}{1 - z}} \]

   if -1.3e52 < y < 6.50000000000000001e103

   1. Initial program 74.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 54.5

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

   5. Applied rewrites 54.5%

      \[\leadsto \color{blue}{\frac{t - a}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 41.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{-122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-36}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -1.6e-122) t_1 (if (<= y 1.65e-36) (/ t b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -1.6e-122) {
		tmp = t_1;
	} else if (y <= 1.65e-36) {
		tmp = t / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (1.0d0 - z)
    if (y <= (-1.6d-122)) then
        tmp = t_1
    else if (y <= 1.65d-36) then
        tmp = t / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -1.6e-122) {
		tmp = t_1;
	} else if (y <= 1.65e-36) {
		tmp = t / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -1.6e-122:
		tmp = t_1
	elif y <= 1.65e-36:
		tmp = t / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -1.6e-122)
		tmp = t_1;
	elseif (y <= 1.65e-36)
		tmp = Float64(t / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -1.6e-122)
		tmp = t_1;
	elseif (y <= 1.65e-36)
		tmp = t / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6e-122], t$95$1, If[LessEqual[y, 1.65e-36], N[(t / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6000000000000001e-122 or 1.64999999999999995e-36 < y

   1. Initial program 61.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{x}{1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}} \]

      3. unsub-neg N/A

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

      4. lower--.f64 40.4

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

   5. Applied rewrites 40.4%

      \[\leadsto \color{blue}{\frac{x}{1 - z}} \]

   if -1.6000000000000001e-122 < y < 1.64999999999999995e-36

   1. Initial program 77.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 77.8

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 77.8

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

   4. Applied rewrites 77.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 59.7

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

   7. Applied rewrites 59.7%

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

   8. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 46.5

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

   10. Applied rewrites 46.5%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 40.5%

       \[\leadsto \frac{t}{\color{blue}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 35.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-55}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-129}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.45e-55) (/ t b) (if (<= z 7.8e-129) (* 1.0 x) (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.45e-55) {
		tmp = t / b;
	} else if (z <= 7.8e-129) {
		tmp = 1.0 * x;
	} else {
		tmp = t / 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 (z <= (-1.45d-55)) then
        tmp = t / b
    else if (z <= 7.8d-129) then
        tmp = 1.0d0 * x
    else
        tmp = t / 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 (z <= -1.45e-55) {
		tmp = t / b;
	} else if (z <= 7.8e-129) {
		tmp = 1.0 * x;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -1.45e-55:
		tmp = t / b
	elif z <= 7.8e-129:
		tmp = 1.0 * x
	else:
		tmp = t / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.45e-55)
		tmp = Float64(t / b);
	elseif (z <= 7.8e-129)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(t / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -1.45e-55)
		tmp = t / b;
	elseif (z <= 7.8e-129)
		tmp = 1.0 * x;
	else
		tmp = t / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.45e-55], N[(t / b), $MachinePrecision], If[LessEqual[z, 7.8e-129], N[(1.0 * x), $MachinePrecision], N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.45e-55 or 7.80000000000000019e-129 < z

   1. Initial program 54.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 54.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 54.6

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

   4. Applied rewrites 54.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 41.9

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

   7. Applied rewrites 41.9%

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

   8. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 36.1

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

   10. Applied rewrites 36.1%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 27.2%

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

   if -1.45e-55 < z < 7.80000000000000019e-129

   1. Initial program 92.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 92.2

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 92.2

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

   4. Applied rewrites 92.2%

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

   5. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 58.4

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

   7. Applied rewrites 58.4%

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

   8. Taylor expanded in z around 0

      \[\leadsto x \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 49.8%

       \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-55}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{-129}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 25.5% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{x}{1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}} \]

   3. unsub-neg N/A

      \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

   4. lower--.f64 28.4

      \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

5. Applied rewrites 28.4%

   \[\leadsto \color{blue}{\frac{x}{1 - z}} \]

6. Taylor expanded in z around 0

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

8. Applied rewrites 21.6%

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

Alternative 15: 25.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 67.3%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 67.4

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 67.4

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

4. Applied rewrites 67.4%

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

5. Taylor expanded in x around inf

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 31.2

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

7. Applied rewrites 31.2%

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

8. Taylor expanded in z around 0

   \[\leadsto x \cdot 1 \]
9. Step-by-step derivation

10. Applied rewrites 21.4%

    \[\leadsto x \cdot 1 \]

11. Final simplification 21.4%

    \[\leadsto 1 \cdot x \]
12. Add Preprocessing

Alternative 16: 3.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ x \cdot z \end{array} \]

FPCore

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

C

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

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 * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{x}{1 + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}} \]

   3. unsub-neg N/A

      \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

   4. lower--.f64 28.4

      \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

5. Applied rewrites 28.4%

   \[\leadsto \color{blue}{\frac{x}{1 - z}} \]

6. Taylor expanded in z around 0

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

8. Applied rewrites 21.6%

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

9. Taylor expanded in z around inf

   \[\leadsto x \cdot z \]
10. Step-by-step derivation

11. Applied rewrites 3.5%

    \[\leadsto z \cdot x \]

12. Final simplification 3.5%

    \[\leadsto x \cdot z \]
13. Add Preprocessing

Developer Target 1: 73.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024248 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z)))))

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