Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.5% → 83.8%

Time: 16.3s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ t_2 := \mathsf{fma}\left(t - a, z, y \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+274}:\\ \;\;\;\;\frac{x}{1 - z} - \frac{\mathsf{fma}\left(t - a, \frac{z}{z - 1}, \frac{\left(z \cdot x\right) \cdot b}{\left(z - 1\right) \cdot \left(z - 1\right)}\right)}{y}\\ \mathbf{elif}\;t\_1 \leq 10^{+284}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{1}{t\_2} \cdot \left(b - y\right), z, \frac{y}{t\_2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* (- t a) z) (* y x)) (+ (* (- b y) z) y)))
        (t_2 (fma (- t a) z (* y x))))
   (if (<= t_1 -1e+274)
     (-
      (/ x (- 1.0 z))
      (/
       (fma (- t a) (/ z (- z 1.0)) (/ (* (* z x) b) (* (- z 1.0) (- z 1.0))))
       y))
     (if (<= t_1 1e+284)
       (/ 1.0 (fma (* (/ 1.0 t_2) (- b y)) z (/ y t_2)))
       (/ (- t a) (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	double t_2 = fma((t - a), z, (y * x));
	double tmp;
	if (t_1 <= -1e+274) {
		tmp = (x / (1.0 - z)) - (fma((t - a), (z / (z - 1.0)), (((z * x) * b) / ((z - 1.0) * (z - 1.0)))) / y);
	} else if (t_1 <= 1e+284) {
		tmp = 1.0 / fma(((1.0 / t_2) * (b - y)), z, (y / t_2));
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(t - a) * z) + Float64(y * x)) / Float64(Float64(Float64(b - y) * z) + y))
	t_2 = fma(Float64(t - a), z, Float64(y * x))
	tmp = 0.0
	if (t_1 <= -1e+274)
		tmp = Float64(Float64(x / Float64(1.0 - z)) - Float64(fma(Float64(t - a), Float64(z / Float64(z - 1.0)), Float64(Float64(Float64(z * x) * b) / Float64(Float64(z - 1.0) * Float64(z - 1.0)))) / y));
	elseif (t_1 <= 1e+284)
		tmp = Float64(1.0 / fma(Float64(Float64(1.0 / t_2) * Float64(b - y)), z, Float64(y / t_2)));
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+274], N[(N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(t - a), $MachinePrecision] * N[(z / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(z * x), $MachinePrecision] * b), $MachinePrecision] / N[(N[(z - 1.0), $MachinePrecision] * N[(z - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+284], N[(1.0 / N[(N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(b - y), $MachinePrecision]), $MachinePrecision] * z + N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.99999999999999921e273

   1. Initial program 34.2%

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

   3. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

   5. Applied rewrites 68.2%

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

   if -9.99999999999999921e273 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.00000000000000008e284

   1. Initial program 92.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 91.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 96.0%

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

   if 1.00000000000000008e284 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 13.6%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 80.2

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

   5. Applied rewrites 80.2%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq -1 \cdot 10^{+274}:\\ \;\;\;\;\frac{x}{1 - z} - \frac{\mathsf{fma}\left(t - a, \frac{z}{z - 1}, \frac{\left(z \cdot x\right) \cdot b}{\left(z - 1\right) \cdot \left(z - 1\right)}\right)}{y}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq 10^{+284}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(t - a, z, y \cdot x\right)} \cdot \left(b - y\right), z, \frac{y}{\mathsf{fma}\left(t - a, z, y \cdot x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(b - y\right) \cdot z + y\\ t_2 := \frac{\left(t - a\right) \cdot z + y \cdot x}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \left(t - a\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-318}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, \mathsf{fma}\left(z, -a, y \cdot x\right)\right)}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(t - a, z, y \cdot x\right)} \cdot \left(b - y\right), z, \frac{1}{x}\right)}\\ \mathbf{elif}\;t\_2 \leq 10^{+284}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* (- b y) z) y)) (t_2 (/ (+ (* (- t a) z) (* y x)) t_1)))
   (if (<= t_2 (- INFINITY))
     (* (/ z (fma (- b y) z y)) (- t a))
     (if (<= t_2 -2e-318)
       (/ (fma z t (fma z (- a) (* y x))) t_1)
       (if (<= t_2 0.0)
         (/ 1.0 (fma (* (/ 1.0 (fma (- t a) z (* y x))) (- b y)) z (/ 1.0 x)))
         (if (<= t_2 1e+284) t_2 (/ (- t a) (- b y))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((b - y) * z) + y;
	double t_2 = (((t - a) * z) + (y * x)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (z / fma((b - y), z, y)) * (t - a);
	} else if (t_2 <= -2e-318) {
		tmp = fma(z, t, fma(z, -a, (y * x))) / t_1;
	} else if (t_2 <= 0.0) {
		tmp = 1.0 / fma(((1.0 / fma((t - a), z, (y * x))) * (b - y)), z, (1.0 / x));
	} else if (t_2 <= 1e+284) {
		tmp = t_2;
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(b - y) * z) + y)
	t_2 = Float64(Float64(Float64(Float64(t - a) * z) + Float64(y * x)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(z / fma(Float64(b - y), z, y)) * Float64(t - a));
	elseif (t_2 <= -2e-318)
		tmp = Float64(fma(z, t, fma(z, Float64(-a), Float64(y * x))) / t_1);
	elseif (t_2 <= 0.0)
		tmp = Float64(1.0 / fma(Float64(Float64(1.0 / fma(Float64(t - a), z, Float64(y * x))) * Float64(b - y)), z, Float64(1.0 / x)));
	elseif (t_2 <= 1e+284)
		tmp = t_2;
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(z / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-318], N[(N[(z * t + N[(z * (-a) + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(1.0 / N[(N[(N[(1.0 / N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b - y), $MachinePrecision]), $MachinePrecision] * z + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+284], t$95$2, N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 26.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 53.7

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

   5. Applied rewrites 53.7%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -2.0000024e-318

   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. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 99.6

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if -2.0000024e-318 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 31.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 31.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 97.5%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 86.9

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

   9. Applied rewrites 86.9%

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

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.00000000000000008e284

   1. Initial program 99.7%

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

   if 1.00000000000000008e284 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 13.6%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 80.2

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

   5. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq -\infty:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \left(t - a\right)\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq -2 \cdot 10^{-318}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, \mathsf{fma}\left(z, -a, y \cdot x\right)\right)}{\left(b - y\right) \cdot z + y}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(t - a, z, y \cdot x\right)} \cdot \left(b - y\right), z, \frac{1}{x}\right)}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq 10^{+284}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - a, z, y \cdot x\right)\\ t_2 := \frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+290}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \left(t - a\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+284}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{1}{t\_1} \cdot \left(b - y\right), z, \frac{y}{t\_1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- t a) z (* y x)))
        (t_2 (/ (+ (* (- t a) z) (* y x)) (+ (* (- b y) z) y))))
   (if (<= t_2 -5e+290)
     (* (/ z (fma (- b y) z y)) (- t a))
     (if (<= t_2 1e+284)
       (/ 1.0 (fma (* (/ 1.0 t_1) (- b y)) z (/ y t_1)))
       (/ (- t a) (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((t - a), z, (y * x));
	double t_2 = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	double tmp;
	if (t_2 <= -5e+290) {
		tmp = (z / fma((b - y), z, y)) * (t - a);
	} else if (t_2 <= 1e+284) {
		tmp = 1.0 / fma(((1.0 / t_1) * (b - y)), z, (y / t_1));
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(t - a), z, Float64(y * x))
	t_2 = Float64(Float64(Float64(Float64(t - a) * z) + Float64(y * x)) / Float64(Float64(Float64(b - y) * z) + y))
	tmp = 0.0
	if (t_2 <= -5e+290)
		tmp = Float64(Float64(z / fma(Float64(b - y), z, y)) * Float64(t - a));
	elseif (t_2 <= 1e+284)
		tmp = Float64(1.0 / fma(Float64(Float64(1.0 / t_1) * Float64(b - y)), z, Float64(y / t_1)));
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+290], N[(N[(z / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+284], N[(1.0 / N[(N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(b - y), $MachinePrecision]), $MachinePrecision] * z + N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -4.9999999999999998e290

   1. Initial program 28.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 55.0

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

   5. Applied rewrites 55.0%

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

   if -4.9999999999999998e290 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.00000000000000008e284

   1. Initial program 92.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 91.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-+.f64 N/A

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

   6. Applied rewrites 96.1%

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

   if 1.00000000000000008e284 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 13.6%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 80.2

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

   5. Applied rewrites 80.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq -5 \cdot 10^{+290}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(b - y, z, y\right)} \cdot \left(t - a\right)\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq 10^{+284}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(t - a, z, y \cdot x\right)} \cdot \left(b - y\right), z, \frac{y}{\mathsf{fma}\left(t - a, z, y \cdot x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, \mathsf{fma}\left(z, -a, y \cdot x\right)\right)}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -9.2e+30)
     t_1
     (if (<= z 7.8e+126)
       (/ (fma z t (fma z (- a) (* y x))) (+ (* (- b y) z) y))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -9.2e+30) {
		tmp = t_1;
	} else if (z <= 7.8e+126) {
		tmp = fma(z, t, fma(z, -a, (y * x))) / (((b - y) * z) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -9.2e+30)
		tmp = t_1;
	elseif (z <= 7.8e+126)
		tmp = Float64(fma(z, t, fma(z, Float64(-a), Float64(y * x))) / Float64(Float64(Float64(b - y) * z) + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.2e+30], t$95$1, If[LessEqual[z, 7.8e+126], N[(N[(z * t + N[(z * (-a) + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.2e30 or 7.79999999999999986e126 < z

   1. Initial program 39.8%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 84.5

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

   5. Applied rewrites 84.5%

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

   if -9.2e30 < z < 7.79999999999999986e126

   1. Initial program 86.1%

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

      1. lift-+.f64 N/A

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

      2. +-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. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 86.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 86.1

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

   4. Applied rewrites 86.1%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, \mathsf{fma}\left(z, -a, y \cdot x\right)\right)}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -9.2e+30)
     t_1
     (if (<= z 7.8e+126)
       (/ (+ (* (- t a) z) (* y x)) (+ (* (- b y) z) y))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -9.2e+30) {
		tmp = t_1;
	} else if (z <= 7.8e+126) {
		tmp = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	} 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 <= (-9.2d+30)) then
        tmp = t_1
    else if (z <= 7.8d+126) then
        tmp = (((t - a) * z) + (y * x)) / (((b - y) * z) + y)
    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 <= -9.2e+30) {
		tmp = t_1;
	} else if (z <= 7.8e+126) {
		tmp = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	} 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 <= -9.2e+30:
		tmp = t_1
	elif z <= 7.8e+126:
		tmp = (((t - a) * z) + (y * x)) / (((b - y) * z) + y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -9.2e+30)
		tmp = t_1;
	elseif (z <= 7.8e+126)
		tmp = Float64(Float64(Float64(Float64(t - a) * z) + Float64(y * x)) / Float64(Float64(Float64(b - y) * z) + y));
	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 <= -9.2e+30)
		tmp = t_1;
	elseif (z <= 7.8e+126)
		tmp = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	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, -9.2e+30], t$95$1, If[LessEqual[z, 7.8e+126], N[(N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.2e30 or 7.79999999999999986e126 < z

   1. Initial program 39.8%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 84.5

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

   5. Applied rewrites 84.5%

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

   if -9.2e30 < z < 7.79999999999999986e126

   1. Initial program 86.1%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.45 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-134}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{t}{y} + x\right) - \frac{a}{y}, z, x\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.45e-13)
     t_1
     (if (<= z -1.35e-134)
       (fma (- (+ (/ t y) x) (/ a y)) z x)
       (if (<= z 4e+39) (/ (fma t z (* y x)) (fma (- b y) z y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.45e-13) {
		tmp = t_1;
	} else if (z <= -1.35e-134) {
		tmp = fma((((t / y) + x) - (a / y)), z, x);
	} else if (z <= 4e+39) {
		tmp = fma(t, z, (y * x)) / fma((b - y), z, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.45e-13)
		tmp = t_1;
	elseif (z <= -1.35e-134)
		tmp = fma(Float64(Float64(Float64(t / y) + x) - Float64(a / y)), z, x);
	elseif (z <= 4e+39)
		tmp = Float64(fma(t, z, Float64(y * x)) / fma(Float64(b - y), z, y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.45e-13], t$95$1, If[LessEqual[z, -1.35e-134], N[(N[(N[(N[(t / y), $MachinePrecision] + x), $MachinePrecision] - N[(a / y), $MachinePrecision]), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[z, 4e+39], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4499999999999999e-13 or 3.99999999999999976e39 < z

   1. Initial program 50.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 79.6

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

   5. Applied rewrites 79.6%

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

   if -1.4499999999999999e-13 < z < -1.3499999999999999e-134

   1. Initial program 82.9%

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

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

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 54.1

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

   5. Applied rewrites 54.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 67.4%

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

   if -1.3499999999999999e-134 < z < 3.99999999999999976e39

   1. Initial program 88.7%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 77.0

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

   5. Applied rewrites 77.0%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{-13}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-134}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{t}{y} + x\right) - \frac{a}{y}, z, x\right)\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.08 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{y} \cdot \mathsf{fma}\left(t - a, z, y \cdot x\right)\\ \mathbf{elif}\;z \leq 1.76 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.08e-16)
     t_1
     (if (<= z -8.5e-102)
       (* (/ 1.0 y) (fma (- t a) z (* y x)))
       (if (<= z 1.76e-15) (* (/ y (fma (- b y) z y)) x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.08e-16) {
		tmp = t_1;
	} else if (z <= -8.5e-102) {
		tmp = (1.0 / y) * fma((t - a), z, (y * x));
	} else if (z <= 1.76e-15) {
		tmp = (y / fma((b - y), z, y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.08e-16)
		tmp = t_1;
	elseif (z <= -8.5e-102)
		tmp = Float64(Float64(1.0 / y) * fma(Float64(t - a), z, Float64(y * x)));
	elseif (z <= 1.76e-15)
		tmp = Float64(Float64(y / fma(Float64(b - y), z, y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.08e-16], t$95$1, If[LessEqual[z, -8.5e-102], N[(N[(1.0 / y), $MachinePrecision] * N[(N[(t - a), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.76e-15], N[(N[(y / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.08e-16 or 1.76e-15 < z

   1. Initial program 53.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 77.9

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

   5. Applied rewrites 77.9%

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

   if -1.08e-16 < z < -8.49999999999999973e-102

   1. Initial program 95.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-+.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. clear-num N/A

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

      16. flip-+ N/A

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

      17. lift-+.f64 N/A

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 63.6

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

   7. Applied rewrites 63.6%

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

   if -8.49999999999999973e-102 < z < 1.76e-15

   1. Initial program 84.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 72.6

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

   5. Applied rewrites 72.6%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{-16}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{1}{y} \cdot \mathsf{fma}\left(t - a, z, y \cdot x\right)\\ \mathbf{elif}\;z \leq 1.76 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4.5e-13)
     t_1
     (if (<= z 4e+39) (/ (fma t z (* y x)) (fma (- b y) z y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -4.5e-13) {
		tmp = t_1;
	} else if (z <= 4e+39) {
		tmp = fma(t, z, (y * x)) / fma((b - y), z, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -4.5e-13)
		tmp = t_1;
	elseif (z <= 4e+39)
		tmp = Float64(fma(t, z, Float64(y * x)) / fma(Float64(b - y), z, y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.5e-13], t$95$1, If[LessEqual[z, 4e+39], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.5e-13 or 3.99999999999999976e39 < z

   1. Initial program 50.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 79.6

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

   5. Applied rewrites 79.6%

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

   if -4.5e-13 < z < 3.99999999999999976e39

   1. Initial program 87.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 71.3

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

   5. Applied rewrites 71.3%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.76 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.05e-16)
     t_1
     (if (<= z 1.76e-15) (* (/ y (fma (- b y) z y)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.05e-16) {
		tmp = t_1;
	} else if (z <= 1.76e-15) {
		tmp = (y / fma((b - y), z, y)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.05e-16)
		tmp = t_1;
	elseif (z <= 1.76e-15)
		tmp = Float64(Float64(y / fma(Float64(b - y), z, y)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.05e-16], t$95$1, If[LessEqual[z, 1.76e-15], N[(N[(y / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0500000000000001e-16 or 1.76e-15 < z

   1. Initial program 53.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 77.9

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

   5. Applied rewrites 77.9%

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

   if -1.0500000000000001e-16 < z < 1.76e-15

   1. Initial program 86.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 67.4

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

   5. Applied rewrites 67.4%

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

Alternative 10: 63.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.3e-17) t_1 (if (<= z 1.6e-15) (fma x z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.3e-17) {
		tmp = t_1;
	} else if (z <= 1.6e-15) {
		tmp = fma(x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.3e-17)
		tmp = t_1;
	elseif (z <= 1.6e-15)
		tmp = fma(x, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.3e-17], t$95$1, If[LessEqual[z, 1.6e-15], N[(x * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.30000000000000002e-17 or 1.6e-15 < z

   1. Initial program 53.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 77.9

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

   5. Applied rewrites 77.9%

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

   if -1.30000000000000002e-17 < z < 1.6e-15

   1. Initial program 86.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 55.3%

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

Alternative 11: 53.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-48}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -5.5e-55) t_1 (if (<= y 1.15e-48) (/ (- 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 <= -5.5e-55) {
		tmp = t_1;
	} else if (y <= 1.15e-48) {
		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 <= (-5.5d-55)) then
        tmp = t_1
    else if (y <= 1.15d-48) 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 <= -5.5e-55) {
		tmp = t_1;
	} else if (y <= 1.15e-48) {
		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 <= -5.5e-55:
		tmp = t_1
	elif y <= 1.15e-48:
		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 <= -5.5e-55)
		tmp = t_1;
	elseif (y <= 1.15e-48)
		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 <= -5.5e-55)
		tmp = t_1;
	elseif (y <= 1.15e-48)
		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, -5.5e-55], t$95$1, If[LessEqual[y, 1.15e-48], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4999999999999999e-55 or 1.15e-48 < y

   1. Initial program 60.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 53.5

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

   5. Applied rewrites 53.5%

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

   if -5.4999999999999999e-55 < y < 1.15e-48

   1. Initial program 82.2%

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

   3. Taylor expanded in y around 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 60.8

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

   5. Applied rewrites 60.8%

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

Alternative 12: 32.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{-z}\\ \mathbf{if}\;z \leq -3700:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- z))))
   (if (<= z -3700.0) t_1 (if (<= z 1.65e+29) (fma x z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / -z;
	double tmp;
	if (z <= -3700.0) {
		tmp = t_1;
	} else if (z <= 1.65e+29) {
		tmp = fma(x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(-z))
	tmp = 0.0
	if (z <= -3700.0)
		tmp = t_1;
	elseif (z <= 1.65e+29)
		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[(x / (-z)), $MachinePrecision]}, If[LessEqual[z, -3700.0], t$95$1, If[LessEqual[z, 1.65e+29], N[(x * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3700 or 1.64999999999999992e29 < z

   1. Initial program 49.5%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 20.0

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 19.5%

      \[\leadsto \frac{x}{-z} \]

   if -3700 < z < 1.64999999999999992e29

   1. Initial program 87.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 51.0

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

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 51.1%

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

Alternative 13: 33.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 - z} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return x / (1.0 - 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 / (1.0d0 - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.8%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. 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 36.6

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

5. Applied rewrites 36.6%

   \[\leadsto \color{blue}{\frac{x}{1 - z}} \]
6. Add Preprocessing

Alternative 14: 26.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.8%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. 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 36.6

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

5. Applied rewrites 36.6%

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

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 29.4%

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

Alternative 15: 25.8% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.8%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. 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 36.6

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

5. Applied rewrites 36.6%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 29.0%

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

Alternative 16: 25.6% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.8%

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

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

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

   9. lower--.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 45.3

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

5. Applied rewrites 45.3%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 35.0%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 29.0%

    \[\leadsto x \cdot 1 \]

12. Final simplification 29.0%

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

Alternative 17: 3.9% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.8%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. 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 36.6

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

5. Applied rewrites 36.6%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 29.0%

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

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. Final simplification 4.0%

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

Developer Target 1: 73.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024236 
(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)))))