Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 67.5% → 95.3%

Time: 14.0s

Alternatives: 15

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: 67.5% 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: 95.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(b - y\right)\\ t_2 := x \cdot y + z \cdot \left(t - a\right)\\ t_3 := \mathsf{fma}\left(z, b - y, y\right)\\ t_4 := y \cdot \left(t - a\right)\\ t_5 := \frac{t\_2}{\mathsf{fma}\left(z, b, \mathsf{fma}\left(z, -y, y\right)\right)}\\ t_6 := \left(b - y\right) \cdot \left(b - y\right)\\ t_7 := \frac{t\_2}{y + t\_1}\\ t_8 := \mathsf{fma}\left(z, \frac{t - a}{t\_3}, x \cdot \frac{y}{t\_3}\right)\\ \mathbf{if}\;t\_7 \leq -\infty:\\ \;\;\;\;t\_8\\ \mathbf{elif}\;t\_7 \leq -1 \cdot 10^{-262}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_7 \leq 0:\\ \;\;\;\;\frac{t - a}{b - y} + \frac{\mathsf{fma}\left(y, \frac{\frac{t\_4}{t\_6} + x \cdot \frac{y}{y - b}}{t\_1}, x \cdot \frac{y}{b - y}\right) + \frac{t\_4}{\left(b - y\right) \cdot \left(y - b\right)}}{z}\\ \mathbf{elif}\;t\_7 \leq 10^{+308}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_7 \leq \infty:\\ \;\;\;\;t\_8\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{t\_1}, \frac{t}{b - y}\right) - \mathsf{fma}\left(y, \frac{t - a}{z \cdot t\_6}, \frac{a}{b - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (- b y)))
        (t_2 (+ (* x y) (* z (- t a))))
        (t_3 (fma z (- b y) y))
        (t_4 (* y (- t a)))
        (t_5 (/ t_2 (fma z b (fma z (- y) y))))
        (t_6 (* (- b y) (- b y)))
        (t_7 (/ t_2 (+ y t_1)))
        (t_8 (fma z (/ (- t a) t_3) (* x (/ y t_3)))))
   (if (<= t_7 (- INFINITY))
     t_8
     (if (<= t_7 -1e-262)
       t_5
       (if (<= t_7 0.0)
         (+
          (/ (- t a) (- b y))
          (/
           (+
            (fma
             y
             (/ (+ (/ t_4 t_6) (* x (/ y (- y b)))) t_1)
             (* x (/ y (- b y))))
            (/ t_4 (* (- b y) (- y b))))
           z))
         (if (<= t_7 1e+308)
           t_5
           (if (<= t_7 INFINITY)
             t_8
             (-
              (fma x (/ y t_1) (/ t (- b y)))
              (fma y (/ (- t a) (* z t_6)) (/ a (- b y)))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (b - y);
	double t_2 = (x * y) + (z * (t - a));
	double t_3 = fma(z, (b - y), y);
	double t_4 = y * (t - a);
	double t_5 = t_2 / fma(z, b, fma(z, -y, y));
	double t_6 = (b - y) * (b - y);
	double t_7 = t_2 / (y + t_1);
	double t_8 = fma(z, ((t - a) / t_3), (x * (y / t_3)));
	double tmp;
	if (t_7 <= -((double) INFINITY)) {
		tmp = t_8;
	} else if (t_7 <= -1e-262) {
		tmp = t_5;
	} else if (t_7 <= 0.0) {
		tmp = ((t - a) / (b - y)) + ((fma(y, (((t_4 / t_6) + (x * (y / (y - b)))) / t_1), (x * (y / (b - y)))) + (t_4 / ((b - y) * (y - b)))) / z);
	} else if (t_7 <= 1e+308) {
		tmp = t_5;
	} else if (t_7 <= ((double) INFINITY)) {
		tmp = t_8;
	} else {
		tmp = fma(x, (y / t_1), (t / (b - y))) - fma(y, ((t - a) / (z * t_6)), (a / (b - y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * Float64(b - y))
	t_2 = Float64(Float64(x * y) + Float64(z * Float64(t - a)))
	t_3 = fma(z, Float64(b - y), y)
	t_4 = Float64(y * Float64(t - a))
	t_5 = Float64(t_2 / fma(z, b, fma(z, Float64(-y), y)))
	t_6 = Float64(Float64(b - y) * Float64(b - y))
	t_7 = Float64(t_2 / Float64(y + t_1))
	t_8 = fma(z, Float64(Float64(t - a) / t_3), Float64(x * Float64(y / t_3)))
	tmp = 0.0
	if (t_7 <= Float64(-Inf))
		tmp = t_8;
	elseif (t_7 <= -1e-262)
		tmp = t_5;
	elseif (t_7 <= 0.0)
		tmp = Float64(Float64(Float64(t - a) / Float64(b - y)) + Float64(Float64(fma(y, Float64(Float64(Float64(t_4 / t_6) + Float64(x * Float64(y / Float64(y - b)))) / t_1), Float64(x * Float64(y / Float64(b - y)))) + Float64(t_4 / Float64(Float64(b - y) * Float64(y - b)))) / z));
	elseif (t_7 <= 1e+308)
		tmp = t_5;
	elseif (t_7 <= Inf)
		tmp = t_8;
	else
		tmp = Float64(fma(x, Float64(y / t_1), Float64(t / Float64(b - y))) - fma(y, Float64(Float64(t - a) / Float64(z * t_6)), Float64(a / Float64(b - y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$4 = N[(y * N[(t - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / N[(z * b + N[(z * (-y) + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(b - y), $MachinePrecision] * N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$2 / N[(y + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(x * N[(y / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$7, (-Infinity)], t$95$8, If[LessEqual[t$95$7, -1e-262], t$95$5, If[LessEqual[t$95$7, 0.0], N[(N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(y * N[(N[(N[(t$95$4 / t$95$6), $MachinePrecision] + N[(x * N[(y / N[(y - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(x * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 / N[(N[(b - y), $MachinePrecision] * N[(y - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$7, 1e+308], t$95$5, If[LessEqual[t$95$7, Infinity], t$95$8, N[(N[(x * N[(y / t$95$1), $MachinePrecision] + N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(N[(t - a), $MachinePrecision] / N[(z * t$95$6), $MachinePrecision]), $MachinePrecision] + N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 1e308 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 20.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000001e-262 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1e308

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if -1.00000000000000001e-262 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 21.9%

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

   3. Taylor expanded in z around -inf

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

   5. Applied rewrites 87.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   8. Applied rewrites 71.3%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-262}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{\mathsf{fma}\left(z, b, \mathsf{fma}\left(z, -y, y\right)\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{t - a}{b - y} + \frac{\mathsf{fma}\left(y, \frac{\frac{y \cdot \left(t - a\right)}{\left(b - y\right) \cdot \left(b - y\right)} + x \cdot \frac{y}{y - b}}{z \cdot \left(b - y\right)}, x \cdot \frac{y}{b - y}\right) + \frac{y \cdot \left(t - a\right)}{\left(b - y\right) \cdot \left(y - b\right)}}{z}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 10^{+308}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{\mathsf{fma}\left(z, b, \mathsf{fma}\left(z, -y, y\right)\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z \cdot \left(b - y\right)}, \frac{t}{b - y}\right) - \mathsf{fma}\left(y, \frac{t - a}{z \cdot \left(\left(b - y\right) \cdot \left(b - y\right)\right)}, \frac{a}{b - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(b - y\right)\\ t_2 := x \cdot y + z \cdot \left(t - a\right)\\ t_3 := \frac{t\_2}{\mathsf{fma}\left(z, b, \mathsf{fma}\left(z, -y, y\right)\right)}\\ t_4 := \mathsf{fma}\left(z, b - y, y\right)\\ t_5 := \frac{t\_2}{y + t\_1}\\ t_6 := \mathsf{fma}\left(z, \frac{t - a}{t\_4}, x \cdot \frac{y}{t\_4}\right)\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 \leq -1 \cdot 10^{-262}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;\frac{\left(t - a\right) + \frac{x \cdot y}{z}}{b}\\ \mathbf{elif}\;t\_5 \leq 10^{+308}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{t\_1}, \frac{t}{b - y}\right) - \mathsf{fma}\left(y, \frac{t - a}{z \cdot \left(\left(b - y\right) \cdot \left(b - y\right)\right)}, \frac{a}{b - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (- b y)))
        (t_2 (+ (* x y) (* z (- t a))))
        (t_3 (/ t_2 (fma z b (fma z (- y) y))))
        (t_4 (fma z (- b y) y))
        (t_5 (/ t_2 (+ y t_1)))
        (t_6 (fma z (/ (- t a) t_4) (* x (/ y t_4)))))
   (if (<= t_5 (- INFINITY))
     t_6
     (if (<= t_5 -1e-262)
       t_3
       (if (<= t_5 0.0)
         (/ (+ (- t a) (/ (* x y) z)) b)
         (if (<= t_5 1e+308)
           t_3
           (if (<= t_5 INFINITY)
             t_6
             (-
              (fma x (/ y t_1) (/ t (- b y)))
              (fma
               y
               (/ (- t a) (* z (* (- b y) (- b y))))
               (/ a (- b y)))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (b - y);
	double t_2 = (x * y) + (z * (t - a));
	double t_3 = t_2 / fma(z, b, fma(z, -y, y));
	double t_4 = fma(z, (b - y), y);
	double t_5 = t_2 / (y + t_1);
	double t_6 = fma(z, ((t - a) / t_4), (x * (y / t_4)));
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = t_6;
	} else if (t_5 <= -1e-262) {
		tmp = t_3;
	} else if (t_5 <= 0.0) {
		tmp = ((t - a) + ((x * y) / z)) / b;
	} else if (t_5 <= 1e+308) {
		tmp = t_3;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = t_6;
	} else {
		tmp = fma(x, (y / t_1), (t / (b - y))) - fma(y, ((t - a) / (z * ((b - y) * (b - y)))), (a / (b - y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * Float64(b - y))
	t_2 = Float64(Float64(x * y) + Float64(z * Float64(t - a)))
	t_3 = Float64(t_2 / fma(z, b, fma(z, Float64(-y), y)))
	t_4 = fma(z, Float64(b - y), y)
	t_5 = Float64(t_2 / Float64(y + t_1))
	t_6 = fma(z, Float64(Float64(t - a) / t_4), Float64(x * Float64(y / t_4)))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = t_6;
	elseif (t_5 <= -1e-262)
		tmp = t_3;
	elseif (t_5 <= 0.0)
		tmp = Float64(Float64(Float64(t - a) + Float64(Float64(x * y) / z)) / b);
	elseif (t_5 <= 1e+308)
		tmp = t_3;
	elseif (t_5 <= Inf)
		tmp = t_6;
	else
		tmp = Float64(fma(x, Float64(y / t_1), Float64(t / Float64(b - y))) - fma(y, Float64(Float64(t - a) / Float64(z * Float64(Float64(b - y) * Float64(b - y)))), Float64(a / Float64(b - y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(z * b + N[(z * (-y) + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / N[(y + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$4), $MachinePrecision] + N[(x * N[(y / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], t$95$6, If[LessEqual[t$95$5, -1e-262], t$95$3, If[LessEqual[t$95$5, 0.0], N[(N[(N[(t - a), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$5, 1e+308], t$95$3, If[LessEqual[t$95$5, Infinity], t$95$6, N[(N[(x * N[(y / t$95$1), $MachinePrecision] + N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(N[(t - a), $MachinePrecision] / N[(z * N[(N[(b - y), $MachinePrecision] * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 1e308 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 20.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000001e-262 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1e308

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if -1.00000000000000001e-262 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 21.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 26.4

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

   5. Applied rewrites 26.4%

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

   6. Taylor expanded in b around -inf

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

   8. Applied rewrites 87.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   8. Applied rewrites 71.3%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-262}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{\mathsf{fma}\left(z, b, \mathsf{fma}\left(z, -y, y\right)\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{\left(t - a\right) + \frac{x \cdot y}{z}}{b}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 10^{+308}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{\mathsf{fma}\left(z, b, \mathsf{fma}\left(z, -y, y\right)\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{z \cdot \left(b - y\right)}, \frac{t}{b - y}\right) - \mathsf{fma}\left(y, \frac{t - a}{z \cdot \left(\left(b - y\right) \cdot \left(b - y\right)\right)}, \frac{a}{b - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y + z \cdot \left(t - a\right)\\ t_2 := \frac{t\_1}{\mathsf{fma}\left(z, b, \mathsf{fma}\left(z, -y, y\right)\right)}\\ t_3 := \mathsf{fma}\left(z, b - y, y\right)\\ t_4 := \frac{t\_1}{y + z \cdot \left(b - y\right)}\\ t_5 := \mathsf{fma}\left(z, \frac{t - a}{t\_3}, x \cdot \frac{y}{t\_3}\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-262}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{\left(t - a\right) + \frac{x \cdot y}{z}}{b}\\ \mathbf{elif}\;t\_4 \leq 10^{+308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* x y) (* z (- t a))))
        (t_2 (/ t_1 (fma z b (fma z (- y) y))))
        (t_3 (fma z (- b y) y))
        (t_4 (/ t_1 (+ y (* z (- b y)))))
        (t_5 (fma z (/ (- t a) t_3) (* x (/ y t_3)))))
   (if (<= t_4 (- INFINITY))
     t_5
     (if (<= t_4 -1e-262)
       t_2
       (if (<= t_4 0.0)
         (/ (+ (- t a) (/ (* x y) z)) b)
         (if (<= t_4 1e+308)
           t_2
           (if (<= t_4 INFINITY) t_5 (/ (- t a) (- b y)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * y) + (z * (t - a));
	double t_2 = t_1 / fma(z, b, fma(z, -y, y));
	double t_3 = fma(z, (b - y), y);
	double t_4 = t_1 / (y + (z * (b - y)));
	double t_5 = fma(z, ((t - a) / t_3), (x * (y / t_3)));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_4 <= -1e-262) {
		tmp = t_2;
	} else if (t_4 <= 0.0) {
		tmp = ((t - a) + ((x * y) / z)) / b;
	} else if (t_4 <= 1e+308) {
		tmp = t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_5;
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * y) + Float64(z * Float64(t - a)))
	t_2 = Float64(t_1 / fma(z, b, fma(z, Float64(-y), y)))
	t_3 = fma(z, Float64(b - y), y)
	t_4 = Float64(t_1 / Float64(y + Float64(z * Float64(b - y))))
	t_5 = fma(z, Float64(Float64(t - a) / t_3), Float64(x * Float64(y / t_3)))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_4 <= -1e-262)
		tmp = t_2;
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(Float64(t - a) + Float64(Float64(x * y) / z)) / b);
	elseif (t_4 <= 1e+308)
		tmp = t_2;
	elseif (t_4 <= Inf)
		tmp = t_5;
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(z * b + N[(z * (-y) + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(x * N[(y / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$5, If[LessEqual[t$95$4, -1e-262], t$95$2, If[LessEqual[t$95$4, 0.0], N[(N[(N[(t - a), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$4, 1e+308], t$95$2, If[LessEqual[t$95$4, Infinity], t$95$5, N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 1e308 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 20.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000001e-262 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1e308

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if -1.00000000000000001e-262 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 21.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 26.4

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

   5. Applied rewrites 26.4%

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

   6. Taylor expanded in b around -inf

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

   8. Applied rewrites 87.6%

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

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

   1. Initial program 0.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 71.2

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

   5. Applied rewrites 71.2%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-262}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{\mathsf{fma}\left(z, b, \mathsf{fma}\left(z, -y, y\right)\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{\left(t - a\right) + \frac{x \cdot y}{z}}{b}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 10^{+308}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{\mathsf{fma}\left(z, b, \mathsf{fma}\left(z, -y, y\right)\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y + z \cdot \left(t - a\right)\\ t_2 := \frac{t\_1}{\mathsf{fma}\left(z, b, \mathsf{fma}\left(z, -y, y\right)\right)}\\ t_3 := \frac{t\_1}{y + z \cdot \left(b - y\right)}\\ t_4 := \mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot 1\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-262}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\left(t - a\right) + \frac{x \cdot y}{z}}{b}\\ \mathbf{elif}\;t\_3 \leq 10^{+308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* x y) (* z (- t a))))
        (t_2 (/ t_1 (fma z b (fma z (- y) y))))
        (t_3 (/ t_1 (+ y (* z (- b y)))))
        (t_4 (fma z (/ (- t a) (fma z (- b y) y)) (* x 1.0))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -1e-262)
       t_2
       (if (<= t_3 0.0)
         (/ (+ (- t a) (/ (* x y) z)) b)
         (if (<= t_3 1e+308)
           t_2
           (if (<= t_3 INFINITY) t_4 (/ (- t a) (- b y)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x * y) + (z * (t - a));
	double t_2 = t_1 / fma(z, b, fma(z, -y, y));
	double t_3 = t_1 / (y + (z * (b - y)));
	double t_4 = fma(z, ((t - a) / fma(z, (b - y), y)), (x * 1.0));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -1e-262) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = ((t - a) + ((x * y) / z)) / b;
	} else if (t_3 <= 1e+308) {
		tmp = t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x * y) + Float64(z * Float64(t - a)))
	t_2 = Float64(t_1 / fma(z, b, fma(z, Float64(-y), y)))
	t_3 = Float64(t_1 / Float64(y + Float64(z * Float64(b - y))))
	t_4 = fma(z, Float64(Float64(t - a) / fma(z, Float64(b - y), y)), Float64(x * 1.0))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -1e-262)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(Float64(t - a) + Float64(Float64(x * y) / z)) / b);
	elseif (t_3 <= 1e+308)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(z * b + N[(z * (-y) + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(N[(t - a), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(x * 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -1e-262], t$95$2, If[LessEqual[t$95$3, 0.0], N[(N[(N[(t - a), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$3, 1e+308], t$95$2, If[LessEqual[t$95$3, Infinity], t$95$4, N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 1e308 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 20.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 79.9%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000001e-262 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1e308

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if -1.00000000000000001e-262 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 21.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 26.4

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

   5. Applied rewrites 26.4%

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

   6. Taylor expanded in b around -inf

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

   8. Applied rewrites 87.6%

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

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

   1. Initial program 0.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 71.2

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

   5. Applied rewrites 71.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot 1\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-262}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{\mathsf{fma}\left(z, b, \mathsf{fma}\left(z, -y, y\right)\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{\left(t - a\right) + \frac{x \cdot y}{z}}{b}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 10^{+308}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{\mathsf{fma}\left(z, b, \mathsf{fma}\left(z, -y, y\right)\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_2 := \mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot 1\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-262}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\left(t - a\right) + \frac{x \cdot y}{z}}{b}\\ \mathbf{elif}\;t\_1 \leq 10^{+308}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_2 (fma z (/ (- t a) (fma z (- b y) y)) (* x 1.0))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -1e-262)
       t_1
       (if (<= t_1 0.0)
         (/ (+ (- t a) (/ (* x y) z)) b)
         (if (<= t_1 1e+308)
           t_1
           (if (<= t_1 INFINITY) t_2 (/ (- t a) (- b y)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_2 = fma(z, ((t - a) / fma(z, (b - y), y)), (x * 1.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -1e-262) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = ((t - a) + ((x * y) / z)) / b;
	} else if (t_1 <= 1e+308) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_2 = fma(z, Float64(Float64(t - a) / fma(z, Float64(b - y), y)), Float64(x * 1.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -1e-262)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(t - a) + Float64(Float64(x * y) / z)) / b);
	elseif (t_1 <= 1e+308)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = 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]}, Block[{t$95$2 = N[(z * N[(N[(t - a), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(x * 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -1e-262], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(N[(t - a), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 1e+308], t$95$1, If[LessEqual[t$95$1, Infinity], t$95$2, N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 1e308 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 20.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 79.9%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000001e-262 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1e308

   1. Initial program 99.5%

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

   if -1.00000000000000001e-262 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 21.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 26.4

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

   5. Applied rewrites 26.4%

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

   6. Taylor expanded in b around -inf

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

   8. Applied rewrites 87.6%

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

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

   1. Initial program 0.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 71.2

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

   5. Applied rewrites 71.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot 1\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-262}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{\left(t - a\right) + \frac{x \cdot y}{z}}{b}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 10^{+308}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+56}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -320000000:\\ \;\;\;\;x \cdot \frac{y}{t\_1}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{t\_1}, x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -2.1e+56)
     t_2
     (if (<= z -320000000.0)
       (* x (/ y t_1))
       (if (<= z 2.6e-11) (fma z (/ (- t a) t_1) (* x 1.0)) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -2.1e+56) {
		tmp = t_2;
	} else if (z <= -320000000.0) {
		tmp = x * (y / t_1);
	} else if (z <= 2.6e-11) {
		tmp = fma(z, ((t - a) / t_1), (x * 1.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -2.1e+56)
		tmp = t_2;
	elseif (z <= -320000000.0)
		tmp = Float64(x * Float64(y / t_1));
	elseif (z <= 2.6e-11)
		tmp = fma(z, Float64(Float64(t - a) / t_1), Float64(x * 1.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.1e+56], t$95$2, If[LessEqual[z, -320000000.0], N[(x * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.6e-11], N[(z * N[(N[(t - a), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(x * 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.10000000000000017e56 or 2.6000000000000001e-11 < z

   1. Initial program 43.3%

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

   3. Taylor expanded in 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 81.2

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

   5. Applied rewrites 81.2%

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

   if -2.10000000000000017e56 < z < -3.2e8

   1. Initial program 46.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{y + z \cdot \left(b - y\right)}} \]
   4. 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. lower-fma.f64 N/A

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

      6. lower--.f64 85.6

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

   5. Applied rewrites 85.6%

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

   if -3.2e8 < z < 2.6000000000000001e-11

   1. Initial program 81.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 88.4

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 74.6%

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

Alternative 7: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.32 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.32e-48)
     t_1
     (if (<= z 1.6e-11) (* x (/ y (fma z (- b y) 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 <= -1.32e-48) {
		tmp = t_1;
	} else if (z <= 1.6e-11) {
		tmp = x * (y / fma(z, (b - y), 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 <= -1.32e-48)
		tmp = t_1;
	elseif (z <= 1.6e-11)
		tmp = Float64(x * Float64(y / fma(z, Float64(b - y), 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, -1.32e-48], t$95$1, If[LessEqual[z, 1.6e-11], N[(x * N[(y / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.32e-48 or 1.59999999999999997e-11 < z

   1. Initial program 47.2%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 77.2

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

   5. Applied rewrites 77.2%

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

   if -1.32e-48 < z < 1.59999999999999997e-11

   1. Initial program 80.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. 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. lower-fma.f64 N/A

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

      6. lower--.f64 68.6

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

   5. Applied rewrites 68.6%

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

Alternative 8: 41.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+215}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-47}:\\ \;\;\;\;\frac{a}{-b}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -2.1e+215)
     t_1
     (if (<= z -4.4e-47) (/ a (- b)) (if (<= z 1.65e-11) (fma x z x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -2.1e+215) {
		tmp = t_1;
	} else if (z <= -4.4e-47) {
		tmp = a / -b;
	} else if (z <= 1.65e-11) {
		tmp = fma(x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -2.1e+215)
		tmp = t_1;
	elseif (z <= -4.4e-47)
		tmp = Float64(a / Float64(-b));
	elseif (z <= 1.65e-11)
		tmp = fma(x, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.1e+215], t$95$1, If[LessEqual[z, -4.4e-47], N[(a / (-b)), $MachinePrecision], If[LessEqual[z, 1.65e-11], N[(x * z + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1000000000000002e215 or 1.6500000000000001e-11 < z

   1. Initial program 42.5%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 26.8

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

   5. Applied rewrites 26.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 49.3%

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

   if -2.1000000000000002e215 < z < -4.40000000000000037e-47

   1. Initial program 55.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 40.4

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

   5. Applied rewrites 40.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 35.8%

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

   if -4.40000000000000037e-47 < z < 1.6500000000000001e-11

   1. Initial program 80.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 57.1

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

   5. Applied rewrites 57.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 57.1%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+215}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-47}:\\ \;\;\;\;\frac{a}{-b}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.32 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.32e-48) t_1 (if (<= z 7.2e-13) (fma x z 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 <= -1.32e-48) {
		tmp = t_1;
	} else if (z <= 7.2e-13) {
		tmp = fma(x, z, 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 <= -1.32e-48)
		tmp = t_1;
	elseif (z <= 7.2e-13)
		tmp = fma(x, z, 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, -1.32e-48], t$95$1, If[LessEqual[z, 7.2e-13], N[(x * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.32e-48 or 7.1999999999999996e-13 < z

   1. Initial program 47.2%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 77.2

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

   5. Applied rewrites 77.2%

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

   if -1.32e-48 < z < 7.1999999999999996e-13

   1. Initial program 80.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 57.1

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

   5. Applied rewrites 57.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 57.1%

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

Alternative 10: 54.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -1.26 \cdot 10^{-70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 250000000000:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -1.26e-70) t_1 (if (<= y 250000000000.0) (/ (- 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 <= -1.26e-70) {
		tmp = t_1;
	} else if (y <= 250000000000.0) {
		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 <= (-1.26d-70)) then
        tmp = t_1
    else if (y <= 250000000000.0d0) 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 <= -1.26e-70) {
		tmp = t_1;
	} else if (y <= 250000000000.0) {
		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 <= -1.26e-70:
		tmp = t_1
	elif y <= 250000000000.0:
		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 <= -1.26e-70)
		tmp = t_1;
	elseif (y <= 250000000000.0)
		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 <= -1.26e-70)
		tmp = t_1;
	elseif (y <= 250000000000.0)
		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, -1.26e-70], t$95$1, If[LessEqual[y, 250000000000.0], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2600000000000001e-70 or 2.5e11 < y

   1. Initial program 53.9%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 -1.2600000000000001e-70 < y < 2.5e11

   1. Initial program 76.5%

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

   3. Taylor expanded in 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 59.7

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

   5. Applied rewrites 59.7%

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

Alternative 11: 42.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -6.35 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -6.35e+147) t_1 (if (<= z 1.65e-11) (/ x (- 1.0 z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -6.35e+147) {
		tmp = t_1;
	} else if (z <= 1.65e-11) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / (b - y)
    if (z <= (-6.35d+147)) then
        tmp = t_1
    else if (z <= 1.65d-11) then
        tmp = x / (1.0d0 - z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -6.35e+147) {
		tmp = t_1;
	} else if (z <= 1.65e-11) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t / (b - y)
	tmp = 0
	if z <= -6.35e+147:
		tmp = t_1
	elif z <= 1.65e-11:
		tmp = x / (1.0 - z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -6.35e+147)
		tmp = t_1;
	elseif (z <= 1.65e-11)
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t / (b - y);
	tmp = 0.0;
	if (z <= -6.35e+147)
		tmp = t_1;
	elseif (z <= 1.65e-11)
		tmp = x / (1.0 - z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.35e+147], t$95$1, If[LessEqual[z, 1.65e-11], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.3500000000000001e147 or 1.6500000000000001e-11 < z

   1. Initial program 41.8%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 25.8

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 46.8%

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

   if -6.3500000000000001e147 < z < 1.6500000000000001e-11

   1. Initial program 76.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 51.1

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

   5. Applied rewrites 51.1%

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

Alternative 12: 36.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{-b}\\ \mathbf{if}\;z \leq -4.4 \cdot 10^{-47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (- b))))
   (if (<= z -4.4e-47) t_1 (if (<= z 1e-12) (fma x z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / -b;
	double tmp;
	if (z <= -4.4e-47) {
		tmp = t_1;
	} else if (z <= 1e-12) {
		tmp = fma(x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(-b))
	tmp = 0.0
	if (z <= -4.4e-47)
		tmp = t_1;
	elseif (z <= 1e-12)
		tmp = fma(x, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / (-b)), $MachinePrecision]}, If[LessEqual[z, -4.4e-47], t$95$1, If[LessEqual[z, 1e-12], N[(x * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.40000000000000037e-47 or 9.9999999999999998e-13 < z

   1. Initial program 47.2%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 27.4

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

   5. Applied rewrites 27.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 33.0%

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

   if -4.40000000000000037e-47 < z < 9.9999999999999998e-13

   1. Initial program 80.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 57.1

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

   5. Applied rewrites 57.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 57.1%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-47}:\\ \;\;\;\;\frac{a}{-b}\\ \mathbf{elif}\;z \leq 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -5.2e-23) (/ t b) (if (<= z 1.65e-11) (fma x z x) (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -5.2e-23) {
		tmp = t / b;
	} else if (z <= 1.65e-11) {
		tmp = fma(x, z, x);
	} else {
		tmp = t / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -5.2e-23)
		tmp = Float64(t / b);
	elseif (z <= 1.65e-11)
		tmp = fma(x, z, x);
	else
		tmp = Float64(t / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -5.2e-23], N[(t / b), $MachinePrecision], If[LessEqual[z, 1.65e-11], N[(x * z + x), $MachinePrecision], N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.2e-23 or 1.6500000000000001e-11 < z

   1. Initial program 46.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 64.0

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in b around -inf

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

   8. Applied rewrites 54.8%

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

   9. Taylor expanded in t around inf

      \[\leadsto \frac{t}{b} \]
   10. Step-by-step derivation

   11. Applied rewrites 25.2%

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

   if -5.2e-23 < z < 1.6500000000000001e-11

   1. Initial program 80.5%

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

   3. Taylor expanded in 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 55.9

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 55.9%

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

Alternative 14: 24.9% 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 63.0%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
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 37.1

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

5. Applied rewrites 37.1%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 29.5%

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

Alternative 15: 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 63.0%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
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 37.1

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

5. Applied rewrites 37.1%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 29.5%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 4.3%

    \[\leadsto z \cdot x \]

12. Final simplification 4.3%

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

Developer Target 1: 74.6% 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 2024238 
(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)))))