Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.6% → 95.0%

Time: 15.3s

Alternatives: 20

Speedup: 0.8×

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: 66.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_2 := \mathsf{fma}\left(z, b - y, y\right)\\ t_3 := \mathsf{fma}\left(z, \frac{t - a}{t\_2}, x \cdot \frac{y}{t\_2}\right)\\ t_4 := \frac{t - a}{b - y}\\ t_5 := \frac{\mathsf{fma}\left(z, t - a, x \cdot y\right)}{t\_2}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+299}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-278}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{x \cdot y}{b - y} + \frac{y \cdot \left(t - a\right)}{\left(b - y\right) \cdot \left(y - b\right)}}{z} + t\_4\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_2 (fma z (- b y) y))
        (t_3 (fma z (/ (- t a) t_2) (* x (/ y t_2))))
        (t_4 (/ (- t a) (- b y)))
        (t_5 (/ (fma z (- t a) (* x y)) t_2)))
   (if (<= t_1 -4e+299)
     t_3
     (if (<= t_1 -1e-278)
       t_5
       (if (<= t_1 0.0)
         (+
          (/ (+ (/ (* x y) (- b y)) (/ (* y (- t a)) (* (- b y) (- y b)))) z)
          t_4)
         (if (<= t_1 2e+306) t_5 (if (<= t_1 INFINITY) t_3 t_4)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_2 = fma(z, (b - y), y);
	double t_3 = fma(z, ((t - a) / t_2), (x * (y / t_2)));
	double t_4 = (t - a) / (b - y);
	double t_5 = fma(z, (t - a), (x * y)) / t_2;
	double tmp;
	if (t_1 <= -4e+299) {
		tmp = t_3;
	} else if (t_1 <= -1e-278) {
		tmp = t_5;
	} else if (t_1 <= 0.0) {
		tmp = ((((x * y) / (b - y)) + ((y * (t - a)) / ((b - y) * (y - b)))) / z) + t_4;
	} else if (t_1 <= 2e+306) {
		tmp = t_5;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_2 = fma(z, Float64(b - y), y)
	t_3 = fma(z, Float64(Float64(t - a) / t_2), Float64(x * Float64(y / t_2)))
	t_4 = Float64(Float64(t - a) / Float64(b - y))
	t_5 = Float64(fma(z, Float64(t - a), Float64(x * y)) / t_2)
	tmp = 0.0
	if (t_1 <= -4e+299)
		tmp = t_3;
	elseif (t_1 <= -1e-278)
		tmp = t_5;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(x * y) / Float64(b - y)) + Float64(Float64(y * Float64(t - a)) / Float64(Float64(b - y) * Float64(y - b)))) / z) + t_4);
	elseif (t_1 <= 2e+306)
		tmp = t_5;
	elseif (t_1 <= Inf)
		tmp = t_3;
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(x * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(z * N[(t - a), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+299], t$95$3, If[LessEqual[t$95$1, -1e-278], t$95$5, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(N[(x * y), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(N[(y * N[(t - a), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * N[(y - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$1, 2e+306], t$95$5, If[LessEqual[t$95$1, Infinity], t$95$3, t$95$4]]]]]]]]]]

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)))) < -4.0000000000000002e299 or 2.00000000000000003e306 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 25.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -4.0000000000000002e299 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.99999999999999938e-279 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 2.00000000000000003e306

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if -9.99999999999999938e-279 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0

   1. Initial program 19.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 25.3

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

   5. Applied rewrites 25.3%

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

   6. Taylor expanded in z around -inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

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

      3. lower-+.f64 N/A

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

   8. Applied rewrites 82.4%

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

   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 74.4

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

   5. Applied rewrites 74.4%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -4 \cdot 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-278}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{\frac{x \cdot y}{b - y} + \frac{y \cdot \left(t - a\right)}{\left(b - y\right) \cdot \left(y - b\right)}}{z} + \frac{t - a}{b - y}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_3 := \mathsf{fma}\left(z, b - y, y\right)\\ t_4 := \mathsf{fma}\left(z, \frac{t - a}{t\_3}, x \cdot \frac{y}{t\_3}\right)\\ t_5 := \frac{\mathsf{fma}\left(z, t - a, x \cdot y\right)}{t\_3}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+299}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-278}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_3 (fma z (- b y) y))
        (t_4 (fma z (/ (- t a) t_3) (* x (/ y t_3))))
        (t_5 (/ (fma z (- t a) (* x y)) t_3)))
   (if (<= t_2 -4e+299)
     t_4
     (if (<= t_2 -1e-278)
       t_5
       (if (<= t_2 0.0)
         t_1
         (if (<= t_2 2e+306) t_5 (if (<= t_2 INFINITY) t_4 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 t_2 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_3 = fma(z, (b - y), y);
	double t_4 = fma(z, ((t - a) / t_3), (x * (y / t_3)));
	double t_5 = fma(z, (t - a), (x * y)) / t_3;
	double tmp;
	if (t_2 <= -4e+299) {
		tmp = t_4;
	} else if (t_2 <= -1e-278) {
		tmp = t_5;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+306) {
		tmp = t_5;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_3 = fma(z, Float64(b - y), y)
	t_4 = fma(z, Float64(Float64(t - a) / t_3), Float64(x * Float64(y / t_3)))
	t_5 = Float64(fma(z, Float64(t - a), Float64(x * y)) / t_3)
	tmp = 0.0
	if (t_2 <= -4e+299)
		tmp = t_4;
	elseif (t_2 <= -1e-278)
		tmp = t_5;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 2e+306)
		tmp = t_5;
	elseif (t_2 <= Inf)
		tmp = t_4;
	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]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(x * N[(y / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(z * N[(t - a), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+299], t$95$4, If[LessEqual[t$95$2, -1e-278], t$95$5, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 2e+306], t$95$5, If[LessEqual[t$95$2, Infinity], t$95$4, t$95$1]]]]]]]]]]

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)))) < -4.0000000000000002e299 or 2.00000000000000003e306 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 25.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -4.0000000000000002e299 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.99999999999999938e-279 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 2.00000000000000003e306

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if -9.99999999999999938e-279 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 6.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 75.7

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

   5. Applied rewrites 75.7%

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

Alternative 3: 90.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_3 := \mathsf{fma}\left(z, b - y, y\right)\\ t_4 := \frac{\mathsf{fma}\left(z, t - a, x \cdot y\right)}{t\_3}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{t\_3}, x \cdot 1\right)\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-278}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{y - y \cdot z}, \frac{x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_3 (fma z (- b y) y))
        (t_4 (/ (fma z (- t a) (* x y)) t_3)))
   (if (<= t_2 -4e+299)
     (fma z (/ (- t a) t_3) (* x 1.0))
     (if (<= t_2 -1e-278)
       t_4
       (if (<= t_2 0.0)
         t_1
         (if (<= t_2 2e+306)
           t_4
           (if (<= t_2 INFINITY)
             (fma z (/ (- t a) (- y (* y z))) (/ x (- 1.0 z)))
             t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double t_2 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_3 = fma(z, (b - y), y);
	double t_4 = fma(z, (t - a), (x * y)) / t_3;
	double tmp;
	if (t_2 <= -4e+299) {
		tmp = fma(z, ((t - a) / t_3), (x * 1.0));
	} else if (t_2 <= -1e-278) {
		tmp = t_4;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+306) {
		tmp = t_4;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = fma(z, ((t - a) / (y - (y * z))), (x / (1.0 - z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_3 = fma(z, Float64(b - y), y)
	t_4 = Float64(fma(z, Float64(t - a), Float64(x * y)) / t_3)
	tmp = 0.0
	if (t_2 <= -4e+299)
		tmp = fma(z, Float64(Float64(t - a) / t_3), Float64(x * 1.0));
	elseif (t_2 <= -1e-278)
		tmp = t_4;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 2e+306)
		tmp = t_4;
	elseif (t_2 <= Inf)
		tmp = fma(z, Float64(Float64(t - a) / Float64(y - Float64(y * z))), Float64(x / Float64(1.0 - z)));
	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]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(z * N[(t - a), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+299], N[(z * N[(N[(t - a), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(x * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -1e-278], t$95$4, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 2e+306], t$95$4, If[LessEqual[t$95$2, Infinity], N[(z * N[(N[(t - a), $MachinePrecision] / N[(y - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]]

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)))) < -4.0000000000000002e299

   1. Initial program 22.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 83.9%

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

   if -4.0000000000000002e299 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.99999999999999938e-279 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 2.00000000000000003e306

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if -9.99999999999999938e-279 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 6.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 75.7

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

   5. Applied rewrites 75.7%

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

   if 2.00000000000000003e306 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 27.5%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 20.3

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

   5. Applied rewrites 20.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 78.1%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -4 \cdot 10^{+299}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot 1\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-278}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{y - y \cdot z}, \frac{x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_3 := \mathsf{fma}\left(z, b - y, y\right)\\ t_4 := \mathsf{fma}\left(z, \frac{t - a}{t\_3}, x \cdot 1\right)\\ t_5 := \frac{\mathsf{fma}\left(z, t - a, x \cdot y\right)}{t\_3}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+299}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-278}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_3 (fma z (- b y) y))
        (t_4 (fma z (/ (- t a) t_3) (* x 1.0)))
        (t_5 (/ (fma z (- t a) (* x y)) t_3)))
   (if (<= t_2 -4e+299)
     t_4
     (if (<= t_2 -1e-278)
       t_5
       (if (<= t_2 0.0)
         t_1
         (if (<= t_2 2e+306) t_5 (if (<= t_2 INFINITY) t_4 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 t_2 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_3 = fma(z, (b - y), y);
	double t_4 = fma(z, ((t - a) / t_3), (x * 1.0));
	double t_5 = fma(z, (t - a), (x * y)) / t_3;
	double tmp;
	if (t_2 <= -4e+299) {
		tmp = t_4;
	} else if (t_2 <= -1e-278) {
		tmp = t_5;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+306) {
		tmp = t_5;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_3 = fma(z, Float64(b - y), y)
	t_4 = fma(z, Float64(Float64(t - a) / t_3), Float64(x * 1.0))
	t_5 = Float64(fma(z, Float64(t - a), Float64(x * y)) / t_3)
	tmp = 0.0
	if (t_2 <= -4e+299)
		tmp = t_4;
	elseif (t_2 <= -1e-278)
		tmp = t_5;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 2e+306)
		tmp = t_5;
	elseif (t_2 <= Inf)
		tmp = t_4;
	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]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(x * 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(z * N[(t - a), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+299], t$95$4, If[LessEqual[t$95$2, -1e-278], t$95$5, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 2e+306], t$95$5, If[LessEqual[t$95$2, Infinity], t$95$4, t$95$1]]]]]]]]]]

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)))) < -4.0000000000000002e299 or 2.00000000000000003e306 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 25.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 76.8%

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

   if -4.0000000000000002e299 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.99999999999999938e-279 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 2.00000000000000003e306

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if -9.99999999999999938e-279 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 6.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 75.7

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

   5. Applied rewrites 75.7%

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

Alternative 5: 68.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := x \cdot \frac{y}{t\_1}\\ t_3 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+40}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-22}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-179}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right)}{t\_1}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-190}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 0.0033:\\ \;\;\;\;\mathsf{fma}\left(z, t - a, x \cdot y\right) \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y))
        (t_2 (* x (/ y t_1)))
        (t_3 (/ (- t a) (- b y))))
   (if (<= z -1e+40)
     t_3
     (if (<= z -2.4e-22)
       t_2
       (if (<= z -4.1e-179)
         (/ (* z (- t a)) t_1)
         (if (<= z 1.8e-190)
           t_2
           (if (<= z 0.0033) (* (fma z (- t a) (* x y)) (/ 1.0 y)) t_3)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = x * (y / t_1);
	double t_3 = (t - a) / (b - y);
	double tmp;
	if (z <= -1e+40) {
		tmp = t_3;
	} else if (z <= -2.4e-22) {
		tmp = t_2;
	} else if (z <= -4.1e-179) {
		tmp = (z * (t - a)) / t_1;
	} else if (z <= 1.8e-190) {
		tmp = t_2;
	} else if (z <= 0.0033) {
		tmp = fma(z, (t - a), (x * y)) * (1.0 / y);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(x * Float64(y / t_1))
	t_3 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1e+40)
		tmp = t_3;
	elseif (z <= -2.4e-22)
		tmp = t_2;
	elseif (z <= -4.1e-179)
		tmp = Float64(Float64(z * Float64(t - a)) / t_1);
	elseif (z <= 1.8e-190)
		tmp = t_2;
	elseif (z <= 0.0033)
		tmp = Float64(fma(z, Float64(t - a), Float64(x * y)) * Float64(1.0 / y));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+40], t$95$3, If[LessEqual[z, -2.4e-22], t$95$2, If[LessEqual[z, -4.1e-179], N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 1.8e-190], t$95$2, If[LessEqual[z, 0.0033], N[(N[(z * N[(t - a), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.00000000000000003e40 or 0.0033 < z

   1. Initial program 42.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 84.1

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

   5. Applied rewrites 84.1%

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

   if -1.00000000000000003e40 < z < -2.40000000000000002e-22 or -4.1e-179 < z < 1.80000000000000003e-190

   1. Initial program 70.1%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 75.6

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

   5. Applied rewrites 75.6%

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

   if -2.40000000000000002e-22 < z < -4.1e-179

   1. Initial program 97.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 73.1

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

   5. Applied rewrites 73.1%

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

   if 1.80000000000000003e-190 < z < 0.0033

   1. Initial program 92.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 76.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 74.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+40}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-179}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-190}:\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 0.0033:\\ \;\;\;\;\mathsf{fma}\left(z, t - a, x \cdot y\right) \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{+16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-184}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot t\right)}{t\_1}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-190}:\\ \;\;\;\;x \cdot \frac{y}{t\_1}\\ \mathbf{elif}\;z \leq 310000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{y - y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -1.25e+16)
     t_2
     (if (<= z -4.2e-184)
       (/ (fma x y (* z t)) t_1)
       (if (<= z 1.8e-190)
         (* x (/ y t_1))
         (if (<= z 310000000.0)
           (/ (fma x y (* z (- t a))) (- y (* y z)))
           t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.25e+16) {
		tmp = t_2;
	} else if (z <= -4.2e-184) {
		tmp = fma(x, y, (z * t)) / t_1;
	} else if (z <= 1.8e-190) {
		tmp = x * (y / t_1);
	} else if (z <= 310000000.0) {
		tmp = fma(x, y, (z * (t - a))) / (y - (y * z));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.25e+16)
		tmp = t_2;
	elseif (z <= -4.2e-184)
		tmp = Float64(fma(x, y, Float64(z * t)) / t_1);
	elseif (z <= 1.8e-190)
		tmp = Float64(x * Float64(y / t_1));
	elseif (z <= 310000000.0)
		tmp = Float64(fma(x, y, Float64(z * Float64(t - a))) / Float64(y - Float64(y * z)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.25e+16], t$95$2, If[LessEqual[z, -4.2e-184], N[(N[(x * y + N[(z * t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 1.8e-190], N[(x * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 310000000.0], N[(N[(x * y + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y - N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.25e16 or 3.1e8 < z

   1. Initial program 41.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if -1.25e16 < z < -4.1999999999999998e-184

   1. Initial program 91.4%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 74.2

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

   5. Applied rewrites 74.2%

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

   if -4.1999999999999998e-184 < z < 1.80000000000000003e-190

   1. Initial program 74.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if 1.80000000000000003e-190 < z < 3.1e8

   1. Initial program 90.1%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+16}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-184}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot t\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-190}:\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 310000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{y - y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{+16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-184}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot t\right)}{t\_1}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-190}:\\ \;\;\;\;x \cdot \frac{y}{t\_1}\\ \mathbf{elif}\;z \leq 0.0033:\\ \;\;\;\;\mathsf{fma}\left(z, t - a, x \cdot y\right) \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -1.25e+16)
     t_2
     (if (<= z -4.2e-184)
       (/ (fma x y (* z t)) t_1)
       (if (<= z 1.8e-190)
         (* x (/ y t_1))
         (if (<= z 0.0033) (* (fma z (- t a) (* x y)) (/ 1.0 y)) t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.25e+16) {
		tmp = t_2;
	} else if (z <= -4.2e-184) {
		tmp = fma(x, y, (z * t)) / t_1;
	} else if (z <= 1.8e-190) {
		tmp = x * (y / t_1);
	} else if (z <= 0.0033) {
		tmp = fma(z, (t - a), (x * y)) * (1.0 / y);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.25e+16)
		tmp = t_2;
	elseif (z <= -4.2e-184)
		tmp = Float64(fma(x, y, Float64(z * t)) / t_1);
	elseif (z <= 1.8e-190)
		tmp = Float64(x * Float64(y / t_1));
	elseif (z <= 0.0033)
		tmp = Float64(fma(z, Float64(t - a), Float64(x * y)) * Float64(1.0 / y));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.25e+16], t$95$2, If[LessEqual[z, -4.2e-184], N[(N[(x * y + N[(z * t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 1.8e-190], N[(x * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.0033], N[(N[(z * N[(t - a), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.25e16 or 0.0033 < z

   1. Initial program 41.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -1.25e16 < z < -4.1999999999999998e-184

   1. Initial program 91.4%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 74.2

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

   5. Applied rewrites 74.2%

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

   if -4.1999999999999998e-184 < z < 1.80000000000000003e-190

   1. Initial program 74.7%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if 1.80000000000000003e-190 < z < 0.0033

   1. Initial program 92.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 76.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 74.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+16}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-184}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot t\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-190}:\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 0.0033:\\ \;\;\;\;\mathsf{fma}\left(z, t - a, x \cdot y\right) \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := x \cdot \frac{y}{t\_1}\\ t_3 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+40}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-22}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-179}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right)}{t\_1}\\ \mathbf{elif}\;z \leq 280000000:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y))
        (t_2 (* x (/ y t_1)))
        (t_3 (/ (- t a) (- b y))))
   (if (<= z -1e+40)
     t_3
     (if (<= z -2.4e-22)
       t_2
       (if (<= z -4.1e-179)
         (/ (* z (- t a)) t_1)
         (if (<= z 280000000.0) t_2 t_3))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = x * (y / t_1);
	double t_3 = (t - a) / (b - y);
	double tmp;
	if (z <= -1e+40) {
		tmp = t_3;
	} else if (z <= -2.4e-22) {
		tmp = t_2;
	} else if (z <= -4.1e-179) {
		tmp = (z * (t - a)) / t_1;
	} else if (z <= 280000000.0) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(x * Float64(y / t_1))
	t_3 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1e+40)
		tmp = t_3;
	elseif (z <= -2.4e-22)
		tmp = t_2;
	elseif (z <= -4.1e-179)
		tmp = Float64(Float64(z * Float64(t - a)) / t_1);
	elseif (z <= 280000000.0)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+40], t$95$3, If[LessEqual[z, -2.4e-22], t$95$2, If[LessEqual[z, -4.1e-179], N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 280000000.0], t$95$2, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.00000000000000003e40 or 2.8e8 < z

   1. Initial program 42.4%

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

   3. Taylor expanded in 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 85.5

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

   5. Applied rewrites 85.5%

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

   if -1.00000000000000003e40 < z < -2.40000000000000002e-22 or -4.1e-179 < z < 2.8e8

   1. Initial program 78.2%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 68.5

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

   5. Applied rewrites 68.5%

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

   if -2.40000000000000002e-22 < z < -4.1e-179

   1. Initial program 97.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 73.1

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

   5. Applied rewrites 73.1%

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

Alternative 9: 65.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := x \cdot \frac{y}{t\_1}\\ t_3 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+40}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-41}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-179}:\\ \;\;\;\;\frac{z \cdot t}{t\_1}\\ \mathbf{elif}\;z \leq 280000000:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y))
        (t_2 (* x (/ y t_1)))
        (t_3 (/ (- t a) (- b y))))
   (if (<= z -1e+40)
     t_3
     (if (<= z -1.8e-41)
       t_2
       (if (<= z -4.1e-179)
         (/ (* z t) t_1)
         (if (<= z 280000000.0) t_2 t_3))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = x * (y / t_1);
	double t_3 = (t - a) / (b - y);
	double tmp;
	if (z <= -1e+40) {
		tmp = t_3;
	} else if (z <= -1.8e-41) {
		tmp = t_2;
	} else if (z <= -4.1e-179) {
		tmp = (z * t) / t_1;
	} else if (z <= 280000000.0) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(x * Float64(y / t_1))
	t_3 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1e+40)
		tmp = t_3;
	elseif (z <= -1.8e-41)
		tmp = t_2;
	elseif (z <= -4.1e-179)
		tmp = Float64(Float64(z * t) / t_1);
	elseif (z <= 280000000.0)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+40], t$95$3, If[LessEqual[z, -1.8e-41], t$95$2, If[LessEqual[z, -4.1e-179], N[(N[(z * t), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 280000000.0], t$95$2, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.00000000000000003e40 or 2.8e8 < z

   1. Initial program 42.4%

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

   3. Taylor expanded in 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 85.5

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

   5. Applied rewrites 85.5%

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

   if -1.00000000000000003e40 < z < -1.8e-41 or -4.1e-179 < z < 2.8e8

   1. Initial program 78.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -1.8e-41 < z < -4.1e-179

   1. Initial program 96.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 56.0

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

   5. Applied rewrites 56.0%

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

Alternative 10: 65.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := x \cdot \frac{y}{t\_1}\\ t_3 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+40}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-41}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-179}:\\ \;\;\;\;t \cdot \frac{z}{t\_1}\\ \mathbf{elif}\;z \leq 280000000:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y))
        (t_2 (* x (/ y t_1)))
        (t_3 (/ (- t a) (- b y))))
   (if (<= z -1e+40)
     t_3
     (if (<= z -1.8e-41)
       t_2
       (if (<= z -4.1e-179)
         (* t (/ z t_1))
         (if (<= z 280000000.0) t_2 t_3))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = x * (y / t_1);
	double t_3 = (t - a) / (b - y);
	double tmp;
	if (z <= -1e+40) {
		tmp = t_3;
	} else if (z <= -1.8e-41) {
		tmp = t_2;
	} else if (z <= -4.1e-179) {
		tmp = t * (z / t_1);
	} else if (z <= 280000000.0) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(x * Float64(y / t_1))
	t_3 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1e+40)
		tmp = t_3;
	elseif (z <= -1.8e-41)
		tmp = t_2;
	elseif (z <= -4.1e-179)
		tmp = Float64(t * Float64(z / t_1));
	elseif (z <= 280000000.0)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+40], t$95$3, If[LessEqual[z, -1.8e-41], t$95$2, If[LessEqual[z, -4.1e-179], N[(t * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 280000000.0], t$95$2, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.00000000000000003e40 or 2.8e8 < z

   1. Initial program 42.4%

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

   3. Taylor expanded in 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 85.5

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

   5. Applied rewrites 85.5%

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

   if -1.00000000000000003e40 < z < -1.8e-41 or -4.1e-179 < z < 2.8e8

   1. Initial program 78.9%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if -1.8e-41 < z < -4.1e-179

   1. Initial program 96.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 55.7

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

   8. Applied rewrites 55.7%

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

Alternative 11: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -8400000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -8400000000000.0)
     t_1
     (if (<= z 7.0) (fma z (/ (- t a) (fma z (- b y) y)) (* x 1.0)) 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 <= -8400000000000.0) {
		tmp = t_1;
	} else if (z <= 7.0) {
		tmp = fma(z, ((t - a) / fma(z, (b - y), y)), (x * 1.0));
	} 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 <= -8400000000000.0)
		tmp = t_1;
	elseif (z <= 7.0)
		tmp = fma(z, Float64(Float64(t - a) / fma(z, Float64(b - y), y)), Float64(x * 1.0));
	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, -8400000000000.0], t$95$1, If[LessEqual[z, 7.0], N[(z * N[(N[(t - a), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(x * 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.4e12 or 7 < z

   1. Initial program 41.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -8.4e12 < z < 7

   1. Initial program 85.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 90.0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 77.5%

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

Alternative 12: 62.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -6 \cdot 10^{-58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-179}:\\ \;\;\;\;t \cdot \frac{z}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 280000000:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -6e-58)
     t_1
     (if (<= z -1.05e-179)
       (* t (/ z (fma z (- b y) y)))
       (if (<= z 280000000.0) (/ x (- 1.0 z)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -6e-58) {
		tmp = t_1;
	} else if (z <= -1.05e-179) {
		tmp = t * (z / fma(z, (b - y), y));
	} else if (z <= 280000000.0) {
		tmp = x / (1.0 - z);
	} 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 <= -6e-58)
		tmp = t_1;
	elseif (z <= -1.05e-179)
		tmp = Float64(t * Float64(z / fma(z, Float64(b - y), y)));
	elseif (z <= 280000000.0)
		tmp = Float64(x / Float64(1.0 - z));
	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, -6e-58], t$95$1, If[LessEqual[z, -1.05e-179], N[(t * N[(z / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 280000000.0], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.00000000000000015e-58 or 2.8e8 < z

   1. Initial program 44.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 80.8

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

   5. Applied rewrites 80.8%

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

   if -6.00000000000000015e-58 < z < -1.0499999999999999e-179

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 50.5

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

   8. Applied rewrites 50.5%

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

   if -1.0499999999999999e-179 < z < 2.8e8

   1. Initial program 82.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 63.4

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

   5. Applied rewrites 63.4%

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

Alternative 13: 63.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{-140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 280000000:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.2e-140) t_1 (if (<= z 280000000.0) (/ x (- 1.0 z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.2e-140) {
		tmp = t_1;
	} else if (z <= 280000000.0) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-3.2d-140)) then
        tmp = t_1
    else if (z <= 280000000.0d0) then
        tmp = x / (1.0d0 - z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.2e-140) {
		tmp = t_1;
	} else if (z <= 280000000.0) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -3.2e-140:
		tmp = t_1
	elif z <= 280000000.0:
		tmp = x / (1.0 - z)
	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 <= -3.2e-140)
		tmp = t_1;
	elseif (z <= 280000000.0)
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -3.2e-140)
		tmp = t_1;
	elseif (z <= 280000000.0)
		tmp = x / (1.0 - z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.2e-140], t$95$1, If[LessEqual[z, 280000000.0], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2000000000000001e-140 or 2.8e8 < z

   1. Initial program 50.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if -3.2000000000000001e-140 < z < 2.8e8

   1. Initial program 83.8%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 58.0

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

   5. Applied rewrites 58.0%

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

Alternative 14: 54.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -0.0037:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{-40}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -0.0037) t_1 (if (<= y 2.75e-40) (/ (- t a) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -0.0037) {
		tmp = t_1;
	} else if (y <= 2.75e-40) {
		tmp = (t - a) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (1.0d0 - z)
    if (y <= (-0.0037d0)) then
        tmp = t_1
    else if (y <= 2.75d-40) then
        tmp = (t - a) / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -0.0037) {
		tmp = t_1;
	} else if (y <= 2.75e-40) {
		tmp = (t - a) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -0.0037:
		tmp = t_1
	elif y <= 2.75e-40:
		tmp = (t - a) / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -0.0037)
		tmp = t_1;
	elseif (y <= 2.75e-40)
		tmp = Float64(Float64(t - a) / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -0.0037)
		tmp = t_1;
	elseif (y <= 2.75e-40)
		tmp = (t - a) / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.0037], t$95$1, If[LessEqual[y, 2.75e-40], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0037000000000000002 or 2.75000000000000001e-40 < y

   1. Initial program 50.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 54.2

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

   5. Applied rewrites 54.2%

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

   if -0.0037000000000000002 < y < 2.75000000000000001e-40

   1. Initial program 76.9%

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

   3. Taylor expanded in y around 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 57.9

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

   5. Applied rewrites 57.9%

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

Alternative 15: 44.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{-140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 58000000000:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -3.2e-140) t_1 (if (<= z 58000000000.0) (/ x (- 1.0 z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -3.2e-140) {
		tmp = t_1;
	} else if (z <= 58000000000.0) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / (b - y)
    if (z <= (-3.2d-140)) then
        tmp = t_1
    else if (z <= 58000000000.0d0) then
        tmp = x / (1.0d0 - z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -3.2e-140) {
		tmp = t_1;
	} else if (z <= 58000000000.0) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t / (b - y)
	tmp = 0
	if z <= -3.2e-140:
		tmp = t_1
	elif z <= 58000000000.0:
		tmp = x / (1.0 - z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -3.2e-140)
		tmp = t_1;
	elseif (z <= 58000000000.0)
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t / (b - y);
	tmp = 0.0;
	if (z <= -3.2e-140)
		tmp = t_1;
	elseif (z <= 58000000000.0)
		tmp = x / (1.0 - z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.2e-140], t$95$1, If[LessEqual[z, 58000000000.0], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2000000000000001e-140 or 5.8e10 < z

   1. Initial program 50.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 44.1%

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

   if -3.2000000000000001e-140 < z < 5.8e10

   1. Initial program 83.8%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 58.0

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

   5. Applied rewrites 58.0%

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

Alternative 16: 44.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{-140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -3.2e-140) t_1 (if (<= z 1e-5) (fma x z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -3.2e-140) {
		tmp = t_1;
	} else if (z <= 1e-5) {
		tmp = fma(x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -3.2e-140)
		tmp = t_1;
	elseif (z <= 1e-5)
		tmp = fma(x, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.2e-140], t$95$1, If[LessEqual[z, 1e-5], N[(x * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.2000000000000001e-140 or 1.00000000000000008e-5 < z

   1. Initial program 50.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 43.3%

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

   if -3.2000000000000001e-140 < z < 1.00000000000000008e-5

   1. Initial program 84.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 57.8

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 57.9%

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

Alternative 17: 37.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\frac{a}{b}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(z, \mathsf{fma}\left(x, z, x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (/ a b))))
   (if (<= z -1e+40) t_1 (if (<= z 1.25e-6) (fma z (fma x z x) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -(a / b);
	double tmp;
	if (z <= -1e+40) {
		tmp = t_1;
	} else if (z <= 1.25e-6) {
		tmp = fma(z, fma(x, z, x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(-Float64(a / b))
	tmp = 0.0
	if (z <= -1e+40)
		tmp = t_1;
	elseif (z <= 1.25e-6)
		tmp = fma(z, fma(x, z, x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = (-N[(a / b), $MachinePrecision])}, If[LessEqual[z, -1e+40], t$95$1, If[LessEqual[z, 1.25e-6], N[(z * N[(x * z + x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.00000000000000003e40 or 1.2500000000000001e-6 < z

   1. Initial program 43.0%

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

   3. Taylor expanded in 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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 18.0

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

   5. Applied rewrites 18.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 22.4%

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

   if -1.00000000000000003e40 < z < 1.2500000000000001e-6

   1. Initial program 83.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 52.6

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

   5. Applied rewrites 52.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 50.5%

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

Alternative 18: 27.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 34.0

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

5. Applied rewrites 34.0%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 28.7%

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

Alternative 19: 26.7% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 34.0

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

5. Applied rewrites 34.0%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 27.5%

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

Alternative 20: 3.9% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.5%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 34.0

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

5. Applied rewrites 34.0%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 27.5%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 4.0%

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

Developer Target 1: 73.7% 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 2024221 
(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)))))