Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.0% → 91.0%

Time: 12.4s

Alternatives: 18

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.0% 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: 91.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := y \cdot \frac{x}{t\_1}\\ t_3 := z \cdot \left(t - a\right)\\ t_4 := \frac{x \cdot y + t\_3}{y + z \cdot \left(b - y\right)}\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+266}:\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, t\_2\right)\\ \mathbf{elif}\;t\_4 \leq 10^{+269}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, t\_3\right)}{t\_1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, \frac{z}{t\_1}, t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z}, y, t - a\right)}{b - y} - \frac{y}{{\left(b - y\right)}^{2}} \cdot \frac{t - a}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y))
        (t_2 (* y (/ x t_1)))
        (t_3 (* z (- t a)))
        (t_4 (/ (+ (* x y) t_3) (+ y (* z (- b y))))))
   (if (<= t_4 -5e+266)
     (fma (- t a) (pow (- b y) -1.0) t_2)
     (if (<= t_4 1e+269)
       (/ (fma x y t_3) t_1)
       (if (<= t_4 INFINITY)
         (fma (- t a) (/ z t_1) t_2)
         (-
          (/ (fma (/ x z) y (- t a)) (- b y))
          (* (/ y (pow (- b y) 2.0)) (/ (- t a) z))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = y * (x / t_1);
	double t_3 = z * (t - a);
	double t_4 = ((x * y) + t_3) / (y + (z * (b - y)));
	double tmp;
	if (t_4 <= -5e+266) {
		tmp = fma((t - a), pow((b - y), -1.0), t_2);
	} else if (t_4 <= 1e+269) {
		tmp = fma(x, y, t_3) / t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = fma((t - a), (z / t_1), t_2);
	} else {
		tmp = (fma((x / z), y, (t - a)) / (b - y)) - ((y / pow((b - y), 2.0)) * ((t - a) / z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(y * Float64(x / t_1))
	t_3 = Float64(z * Float64(t - a))
	t_4 = Float64(Float64(Float64(x * y) + t_3) / Float64(y + Float64(z * Float64(b - y))))
	tmp = 0.0
	if (t_4 <= -5e+266)
		tmp = fma(Float64(t - a), (Float64(b - y) ^ -1.0), t_2);
	elseif (t_4 <= 1e+269)
		tmp = Float64(fma(x, y, t_3) / t_1);
	elseif (t_4 <= Inf)
		tmp = fma(Float64(t - a), Float64(z / t_1), t_2);
	else
		tmp = Float64(Float64(fma(Float64(x / z), y, Float64(t - a)) / Float64(b - y)) - Float64(Float64(y / (Float64(b - y) ^ 2.0)) * Float64(Float64(t - a) / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x * y), $MachinePrecision] + t$95$3), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+266], N[(N[(t - a), $MachinePrecision] * N[Power[N[(b - y), $MachinePrecision], -1.0], $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 1e+269], N[(N[(x * y + t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(t - a), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision], N[(N[(N[(N[(x / z), $MachinePrecision] * y + N[(t - a), $MachinePrecision]), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(N[(y / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t - a), $MachinePrecision] / z), $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)))) < -4.9999999999999999e266

   1. Initial program 27.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if -4.9999999999999999e266 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1e269

   1. Initial program 94.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 80.8

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

   4. Applied rewrites 80.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. sub-div N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 94.5

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

   6. Applied rewrites 94.5%

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

   if 1e269 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 35.4%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   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

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. times-frac N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower--.f64 N/A

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

      14. *-commutative N/A

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

   5. Applied rewrites 91.4%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -5 \cdot 10^{+266}:\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, y \cdot \frac{x}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 10^{+269}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(t - a, \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}, y \cdot \frac{x}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{x}{z}, y, t - a\right)}{b - y} - \frac{y}{{\left(b - y\right)}^{2}} \cdot \frac{t - a}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := z \cdot \left(t - a\right)\\ t_3 := \frac{x \cdot y + t\_2}{y + z \cdot \left(b - y\right)}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+266} \lor \neg \left(t\_3 \leq 10^{+269}\right):\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, y \cdot \frac{x}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, t\_2\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y))
        (t_2 (* z (- t a)))
        (t_3 (/ (+ (* x y) t_2) (+ y (* z (- b y))))))
   (if (or (<= t_3 -5e+266) (not (<= t_3 1e+269)))
     (fma (- t a) (pow (- b y) -1.0) (* y (/ x t_1)))
     (/ (fma x y t_2) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = z * (t - a);
	double t_3 = ((x * y) + t_2) / (y + (z * (b - y)));
	double tmp;
	if ((t_3 <= -5e+266) || !(t_3 <= 1e+269)) {
		tmp = fma((t - a), pow((b - y), -1.0), (y * (x / t_1)));
	} else {
		tmp = fma(x, y, t_2) / t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(z * Float64(t - a))
	t_3 = Float64(Float64(Float64(x * y) + t_2) / Float64(y + Float64(z * Float64(b - y))))
	tmp = 0.0
	if ((t_3 <= -5e+266) || !(t_3 <= 1e+269))
		tmp = fma(Float64(t - a), (Float64(b - y) ^ -1.0), Float64(y * Float64(x / t_1)));
	else
		tmp = Float64(fma(x, y, t_2) / t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + t$95$2), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$3, -5e+266], N[Not[LessEqual[t$95$3, 1e+269]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] * N[Power[N[(b - y), $MachinePrecision], -1.0], $MachinePrecision] + N[(y * N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y + t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -4.9999999999999999e266 or 1e269 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 18.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 87.2

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

   7. Applied rewrites 87.2%

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

   if -4.9999999999999999e266 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1e269

   1. Initial program 94.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 80.8

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

   4. Applied rewrites 80.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. sub-div N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 94.5

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

   6. Applied rewrites 94.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -5 \cdot 10^{+266} \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 10^{+269}\right):\\ \;\;\;\;\mathsf{fma}\left(t - a, {\left(b - y\right)}^{-1}, y \cdot \frac{x}{\mathsf{fma}\left(b - y, z, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{\mathsf{fma}\left(z, \frac{t - a}{x}, y\right)}{t\_1} \cdot x\\ t_3 := z \cdot \left(t - a\right)\\ t_4 := \frac{x \cdot y + t\_3}{y + z \cdot \left(b - y\right)}\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{+253}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, t\_3\right)}{t\_1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y))
        (t_2 (* (/ (fma z (/ (- t a) x) y) t_1) x))
        (t_3 (* z (- t a)))
        (t_4 (/ (+ (* x y) t_3) (+ y (* z (- b y))))))
   (if (<= t_4 -1e+253)
     t_2
     (if (<= t_4 2e+224)
       (/ (fma x y t_3) t_1)
       (if (<= t_4 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 = fma((b - y), z, y);
	double t_2 = (fma(z, ((t - a) / x), y) / t_1) * x;
	double t_3 = z * (t - a);
	double t_4 = ((x * y) + t_3) / (y + (z * (b - y)));
	double tmp;
	if (t_4 <= -1e+253) {
		tmp = t_2;
	} else if (t_4 <= 2e+224) {
		tmp = fma(x, y, t_3) / t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(fma(z, Float64(Float64(t - a) / x), y) / t_1) * x)
	t_3 = Float64(z * Float64(t - a))
	t_4 = Float64(Float64(Float64(x * y) + t_3) / Float64(y + Float64(z * Float64(b - y))))
	tmp = 0.0
	if (t_4 <= -1e+253)
		tmp = t_2;
	elseif (t_4 <= 2e+224)
		tmp = Float64(fma(x, y, t_3) / t_1);
	elseif (t_4 <= 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[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z * N[(N[(t - a), $MachinePrecision] / x), $MachinePrecision] + y), $MachinePrecision] / t$95$1), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x * y), $MachinePrecision] + t$95$3), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1e+253], t$95$2, If[LessEqual[t$95$4, 2e+224], N[(N[(x * y + t$95$3), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$2, N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999994e252 or 1.99999999999999994e224 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 39.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 91.4

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 84.7

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

   7. Applied rewrites 84.7%

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

   if -9.9999999999999994e252 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.99999999999999994e224

   1. Initial program 94.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 82.8

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

   4. Applied rewrites 82.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. sub-div N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 94.3

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

   6. Applied rewrites 94.3%

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

   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 72.2

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

   5. Applied rewrites 72.2%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{+253}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{t - a}{x}, y\right)}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 2 \cdot 10^{+224}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, \frac{t - a}{x}, y\right)}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-136}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{y}, z, x\right)\\ \mathbf{elif}\;z \leq 124:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{x}{z}, t - a\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.6e-136)
     t_1
     (if (<= z 1e-80)
       (fma (/ (- t a) y) z x)
       (if (<= z 124.0) (/ (fma y (/ x z) (- t a)) b) 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 <= -3.6e-136) {
		tmp = t_1;
	} else if (z <= 1e-80) {
		tmp = fma(((t - a) / y), z, x);
	} else if (z <= 124.0) {
		tmp = fma(y, (x / z), (t - a)) / b;
	} 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 <= -3.6e-136)
		tmp = t_1;
	elseif (z <= 1e-80)
		tmp = fma(Float64(Float64(t - a) / y), z, x);
	elseif (z <= 124.0)
		tmp = Float64(fma(y, Float64(x / z), Float64(t - a)) / b);
	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, -3.6e-136], t$95$1, If[LessEqual[z, 1e-80], N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[z, 124.0], N[(N[(y * N[(x / z), $MachinePrecision] + N[(t - a), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.5999999999999998e-136 or 124 < z

   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 z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 75.0

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

   5. Applied rewrites 75.0%

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

   if -3.5999999999999998e-136 < z < 9.99999999999999961e-81

   1. Initial program 87.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 58.2

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.5%

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

   if 9.99999999999999961e-81 < z < 124

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 83.3

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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 61.9

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

   7. Applied rewrites 61.9%

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

   9. Applied rewrites 61.9%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{y}, z, x\right)\\ \mathbf{elif}\;z \leq 124:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \frac{x}{z}, t - a\right)}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.6 \cdot 10^{-136}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{y}, z, x\right)\\ \mathbf{elif}\;z \leq 124:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{y}{z}, t\right) - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.6e-136)
     t_1
     (if (<= z 1e-80)
       (fma (/ (- t a) y) z x)
       (if (<= z 124.0) (/ (- (fma x (/ y z) t) a) b) 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 <= -3.6e-136) {
		tmp = t_1;
	} else if (z <= 1e-80) {
		tmp = fma(((t - a) / y), z, x);
	} else if (z <= 124.0) {
		tmp = (fma(x, (y / z), t) - a) / b;
	} 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 <= -3.6e-136)
		tmp = t_1;
	elseif (z <= 1e-80)
		tmp = fma(Float64(Float64(t - a) / y), z, x);
	elseif (z <= 124.0)
		tmp = Float64(Float64(fma(x, Float64(y / z), t) - a) / b);
	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, -3.6e-136], t$95$1, If[LessEqual[z, 1e-80], N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[z, 124.0], N[(N[(N[(x * N[(y / z), $MachinePrecision] + t), $MachinePrecision] - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.5999999999999998e-136 or 124 < z

   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 z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 75.0

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

   5. Applied rewrites 75.0%

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

   if -3.5999999999999998e-136 < z < 9.99999999999999961e-81

   1. Initial program 87.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 58.2

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.5%

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

   if 9.99999999999999961e-81 < z < 124

   1. Initial program 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 83.3

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

   4. Applied rewrites 83.3%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 61.9

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

   7. Applied rewrites 61.9%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{y}, z, x\right)\\ \mathbf{elif}\;z \leq 124:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \frac{y}{z}, t\right) - a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+117} \lor \neg \left(z \leq 2.95 \cdot 10^{+88}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.1e+117) (not (<= z 2.95e+88)))
   (/ (- t a) (- b y))
   (/ (fma x y (* z (- t a))) (fma (- b y) z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.1e+117) || !(z <= 2.95e+88)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(x, y, (z * (t - a))) / fma((b - y), z, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.1e+117) || !(z <= 2.95e+88))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(fma(x, y, Float64(z * Float64(t - a))) / fma(Float64(b - y), z, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.1e+117], N[Not[LessEqual[z, 2.95e+88]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(x * y + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.10000000000000007e117 or 2.94999999999999984e88 < z

   1. Initial program 31.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 85.4

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

   5. Applied rewrites 85.4%

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

   if -1.10000000000000007e117 < z < 2.94999999999999984e88

   1. Initial program 88.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 84.5

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

   4. Applied rewrites 84.5%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. sub-div N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-/.f64 88.5

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

   6. Applied rewrites 88.5%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+117} \lor \neg \left(z \leq 2.95 \cdot 10^{+88}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+19} \lor \neg \left(z \leq 252\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot t\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.2e+19) (not (<= z 252.0)))
   (/ (- t a) (- b y))
   (/ (fma x y (* z t)) (fma (- b y) z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.2e+19) || !(z <= 252.0)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(x, y, (z * t)) / fma((b - y), z, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.2e+19) || !(z <= 252.0))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(fma(x, y, Float64(z * t)) / fma(Float64(b - y), z, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.2e+19], N[Not[LessEqual[z, 252.0]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(x * y + N[(z * t), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2e19 or 252 < z

   1. Initial program 45.2%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 80.4

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

   5. Applied rewrites 80.4%

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

   if -1.2e19 < z < 252

   1. Initial program 89.6%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

      1. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, y, \mathsf{Rewrite=>}\left(lower-*.f64, \left(z \cdot t\right)\right)\right)}{\color{blue}{y + z \cdot \left(b - y\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, y, \mathsf{Rewrite=>}\left(lower-*.f64, \left(z \cdot t\right)\right)\right)}{y + \color{blue}{z \cdot \left(b - y\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, y, \mathsf{Rewrite=>}\left(lower-*.f64, \left(z \cdot t\right)\right)\right)}{y + \color{blue}{\left(b - y\right) \cdot z}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, y, \mathsf{Rewrite=>}\left(lower-*.f64, \left(z \cdot t\right)\right)\right)}{\color{blue}{\left(b - y\right) \cdot z + y}} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x, y, \mathsf{Rewrite=>}\left(lower-*.f64, \left(z \cdot t\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(b - y, z, y\right)}} \]

   7. Applied rewrites 66.7%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+19} \lor \neg \left(z \leq 252\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot t\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+19} \lor \neg \left(z \leq 252\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.2e+19) (not (<= z 252.0)))
   (/ (- t a) (- b y))
   (/ (fma t z (* y x)) (fma (- b y) z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.2e+19) || !(z <= 252.0)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(t, z, (y * x)) / fma((b - y), z, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.2e+19) || !(z <= 252.0))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(fma(t, z, Float64(y * x)) / fma(Float64(b - y), z, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.2e+19], N[Not[LessEqual[z, 252.0]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2e19 or 252 < z

   1. Initial program 45.2%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 80.4

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

   5. Applied rewrites 80.4%

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

   if -1.2e19 < z < 252

   1. Initial program 89.6%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 66.7

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

   5. Applied rewrites 66.7%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+19} \lor \neg \left(z \leq 252\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-a}{b}\\ t_2 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{-65}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -2.55 \cdot 10^{-276}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a) b)) (t_2 (/ x (- 1.0 z))))
   (if (<= y -3.8e-65)
     t_2
     (if (<= y -2.55e-276)
       t_1
       (if (<= y 5.6e-109) (/ t b) (if (<= y 2.2e-19) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -a / b;
	double t_2 = x / (1.0 - z);
	double tmp;
	if (y <= -3.8e-65) {
		tmp = t_2;
	} else if (y <= -2.55e-276) {
		tmp = t_1;
	} else if (y <= 5.6e-109) {
		tmp = t / b;
	} else if (y <= 2.2e-19) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -a / b
    t_2 = x / (1.0d0 - z)
    if (y <= (-3.8d-65)) then
        tmp = t_2
    else if (y <= (-2.55d-276)) then
        tmp = t_1
    else if (y <= 5.6d-109) then
        tmp = t / b
    else if (y <= 2.2d-19) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -a / b;
	double t_2 = x / (1.0 - z);
	double tmp;
	if (y <= -3.8e-65) {
		tmp = t_2;
	} else if (y <= -2.55e-276) {
		tmp = t_1;
	} else if (y <= 5.6e-109) {
		tmp = t / b;
	} else if (y <= 2.2e-19) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = -a / b
	t_2 = x / (1.0 - z)
	tmp = 0
	if y <= -3.8e-65:
		tmp = t_2
	elif y <= -2.55e-276:
		tmp = t_1
	elif y <= 5.6e-109:
		tmp = t / b
	elif y <= 2.2e-19:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(-a) / b)
	t_2 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -3.8e-65)
		tmp = t_2;
	elseif (y <= -2.55e-276)
		tmp = t_1;
	elseif (y <= 5.6e-109)
		tmp = Float64(t / b);
	elseif (y <= 2.2e-19)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = -a / b;
	t_2 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -3.8e-65)
		tmp = t_2;
	elseif (y <= -2.55e-276)
		tmp = t_1;
	elseif (y <= 5.6e-109)
		tmp = t / b;
	elseif (y <= 2.2e-19)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[((-a) / b), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.8e-65], t$95$2, If[LessEqual[y, -2.55e-276], t$95$1, If[LessEqual[y, 5.6e-109], N[(t / b), $MachinePrecision], If[LessEqual[y, 2.2e-19], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8000000000000002e-65 or 2.1999999999999998e-19 < y

   1. Initial program 64.5%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 53.2

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

   5. Applied rewrites 53.2%

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

   if -3.8000000000000002e-65 < y < -2.54999999999999984e-276 or 5.59999999999999958e-109 < y < 2.1999999999999998e-19

   1. Initial program 76.1%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 48.2

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

   5. Applied rewrites 48.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 47.1%

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

   if -2.54999999999999984e-276 < y < 5.59999999999999958e-109

   1. Initial program 74.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 81.2

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

   4. Applied rewrites 81.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 76.3

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

   7. Applied rewrites 76.3%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 47.9%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;y \leq -2.55 \cdot 10^{-276}:\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-136} \lor \neg \left(z \leq 1.06 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{y}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -3.6e-136) (not (<= z 1.06e-9)))
   (/ (- t a) (- b y))
   (fma (/ (- t a) y) z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -3.6e-136) || !(z <= 1.06e-9)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(((t - a) / y), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -3.6e-136) || !(z <= 1.06e-9))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = fma(Float64(Float64(t - a) / y), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -3.6e-136], N[Not[LessEqual[z, 1.06e-9]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.5999999999999998e-136 or 1.0600000000000001e-9 < z

   1. Initial program 55.2%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 73.8

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

   5. Applied rewrites 73.8%

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

   if -3.5999999999999998e-136 < z < 1.0600000000000001e-9

   1. Initial program 89.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 56.6

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 67.8%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-136} \lor \neg \left(z \leq 1.06 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{y}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.35 \cdot 10^{-136} \lor \neg \left(z \leq 1.1 \cdot 10^{-80}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{y}, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -3.35e-136) (not (<= z 1.1e-80)))
   (/ (- t a) (- b y))
   (fma (/ (- a) y) z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -3.35e-136) || !(z <= 1.1e-80)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma((-a / y), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -3.35e-136) || !(z <= 1.1e-80))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = fma(Float64(Float64(-a) / y), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -3.35e-136], N[Not[LessEqual[z, 1.1e-80]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[((-a) / y), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3499999999999999e-136 or 1.10000000000000005e-80 < z

   1. Initial program 60.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

   if -3.3499999999999999e-136 < z < 1.10000000000000005e-80

   1. Initial program 87.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 58.2

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 63.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.35 \cdot 10^{-136} \lor \neg \left(z \leq 1.1 \cdot 10^{-80}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{y}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-136} \lor \neg \left(z \leq 10^{-80}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.4e-136) (not (<= z 1e-80))) (/ (- t a) (- b y)) (fma x z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.4e-136) || !(z <= 1e-80)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.4e-136) || !(z <= 1e-80))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.4e-136], N[Not[LessEqual[z, 1e-80]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3999999999999999e-136 or 9.99999999999999961e-81 < z

   1. Initial program 60.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 70.3

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

   5. Applied rewrites 70.3%

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

   if -2.3999999999999999e-136 < z < 9.99999999999999961e-81

   1. Initial program 87.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 55.2%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-136} \lor \neg \left(z \leq 10^{-80}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-74} \lor \neg \left(y \leq 1.55 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -4.5e-74) (not (<= y 1.55e-18))) (/ x (- 1.0 z)) (/ (- t a) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -4.5e-74) || !(y <= 1.55e-18)) {
		tmp = x / (1.0 - z);
	} else {
		tmp = (t - a) / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-4.5d-74)) .or. (.not. (y <= 1.55d-18))) then
        tmp = x / (1.0d0 - z)
    else
        tmp = (t - a) / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -4.5e-74) || !(y <= 1.55e-18)) {
		tmp = x / (1.0 - z);
	} else {
		tmp = (t - a) / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -4.5e-74) or not (y <= 1.55e-18):
		tmp = x / (1.0 - z)
	else:
		tmp = (t - a) / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -4.5e-74) || !(y <= 1.55e-18))
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = Float64(Float64(t - a) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -4.5e-74) || ~((y <= 1.55e-18)))
		tmp = x / (1.0 - z);
	else
		tmp = (t - a) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -4.5e-74], N[Not[LessEqual[y, 1.55e-18]], $MachinePrecision]], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.4999999999999999e-74 or 1.55000000000000003e-18 < y

   1. Initial program 64.7%

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

   3. Taylor expanded in y around 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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 52.8

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

   5. Applied rewrites 52.8%

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

   if -4.4999999999999999e-74 < y < 1.55000000000000003e-18

   1. Initial program 75.0%

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

   3. Taylor expanded in y around 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 68.1

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

   5. Applied rewrites 68.1%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-74} \lor \neg \left(y \leq 1.55 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 1.72 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -4.2e-103) (/ t b) (if (<= z 1.72e-65) (/ x 1.0) (/ (- a) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -4.2e-103) {
		tmp = t / b;
	} else if (z <= 1.72e-65) {
		tmp = x / 1.0;
	} else {
		tmp = -a / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-4.2d-103)) then
        tmp = t / b
    else if (z <= 1.72d-65) then
        tmp = x / 1.0d0
    else
        tmp = -a / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -4.2e-103) {
		tmp = t / b;
	} else if (z <= 1.72e-65) {
		tmp = x / 1.0;
	} else {
		tmp = -a / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -4.2e-103:
		tmp = t / b
	elif z <= 1.72e-65:
		tmp = x / 1.0
	else:
		tmp = -a / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -4.2e-103)
		tmp = Float64(t / b);
	elseif (z <= 1.72e-65)
		tmp = Float64(x / 1.0);
	else
		tmp = Float64(Float64(-a) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -4.2e-103)
		tmp = t / b;
	elseif (z <= 1.72e-65)
		tmp = x / 1.0;
	else
		tmp = -a / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -4.2e-103], N[(t / b), $MachinePrecision], If[LessEqual[z, 1.72e-65], N[(x / 1.0), $MachinePrecision], N[((-a) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.20000000000000009e-103

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 70.9

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

   4. Applied rewrites 70.9%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 55.2

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

   7. Applied rewrites 55.2%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 34.8%

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

   if -4.20000000000000009e-103 < z < 1.72000000000000005e-65

   1. Initial program 88.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 54.1

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

   5. Applied rewrites 54.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 54.1%

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

   if 1.72000000000000005e-65 < z

   1. Initial program 63.3%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 47.8

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

   5. Applied rewrites 47.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 40.5%

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

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 1.72 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{-118} \lor \neg \left(y \leq 3.8 \cdot 10^{-105}\right):\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -2.45e-118) (not (<= y 3.8e-105))) (/ x 1.0) (/ t b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.45e-118) || !(y <= 3.8e-105)) {
		tmp = x / 1.0;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-2.45d-118)) .or. (.not. (y <= 3.8d-105))) then
        tmp = x / 1.0d0
    else
        tmp = t / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -2.45e-118) || !(y <= 3.8e-105)) {
		tmp = x / 1.0;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -2.45e-118) or not (y <= 3.8e-105):
		tmp = x / 1.0
	else:
		tmp = t / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -2.45e-118) || !(y <= 3.8e-105))
		tmp = Float64(x / 1.0);
	else
		tmp = Float64(t / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -2.45e-118) || ~((y <= 3.8e-105)))
		tmp = x / 1.0;
	else
		tmp = t / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -2.45e-118], N[Not[LessEqual[y, 3.8e-105]], $MachinePrecision]], N[(x / 1.0), $MachinePrecision], N[(t / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4499999999999999e-118 or 3.7999999999999998e-105 < y

   1. Initial program 67.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 46.4

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 35.6%

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

   if -2.4499999999999999e-118 < y < 3.7999999999999998e-105

   1. Initial program 73.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 81.9

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

   4. Applied rewrites 81.9%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 77.2

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

   7. Applied rewrites 77.2%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 40.7%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{-118} \lor \neg \left(y \leq 3.8 \cdot 10^{-105}\right):\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 36.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-103} \lor \neg \left(z \leq 1.4 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -4.2e-103) (not (<= z 1.4e-8))) (/ t b) (fma x z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -4.2e-103) || !(z <= 1.4e-8)) {
		tmp = t / b;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -4.2e-103) || !(z <= 1.4e-8))
		tmp = Float64(t / b);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -4.2e-103], N[Not[LessEqual[z, 1.4e-8]], $MachinePrecision]], N[(t / b), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.20000000000000009e-103 or 1.4e-8 < z

   1. Initial program 54.2%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-/l* N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 71.4

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 56.8

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

   7. Applied rewrites 56.8%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 27.9%

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

   if -4.20000000000000009e-103 < z < 1.4e-8

   1. Initial program 89.5%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 49.2

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

   5. Applied rewrites 49.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 49.2%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-103} \lor \neg \left(z \leq 1.4 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 25.9% 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 69.8%

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

3. Taylor expanded in 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. fp-cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower--.f64 31.5

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

5. Applied rewrites 31.5%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 24.5%

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

9. Final simplification 24.5%

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

Alternative 18: 3.8% 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 69.8%

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

3. Taylor expanded in 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. fp-cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower--.f64 31.5

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

5. Applied rewrites 31.5%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 24.5%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 3.2%

    \[\leadsto x \cdot z \]

12. Final simplification 3.2%

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

Developer Target 1: 73.2% 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 2024337 
(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)))))