Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.3% → 88.9%

Time: 15.6s

Alternatives: 19

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_3 := \mathsf{fma}\left(z, t - a, x \cdot y\right)\\ t_4 := \frac{z}{t\_3}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+280}:\\ \;\;\;\;\left(-\frac{x}{z + -1}\right) - \frac{\mathsf{fma}\left(z, \frac{t - a}{z + -1}, \frac{z \cdot \left(x \cdot b\right)}{\left(z + -1\right) \cdot \left(z + -1\right)}\right)}{y}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, t\_4, \frac{y}{t\_3}\right) - y \cdot t\_4}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, \frac{t - a}{x \cdot t\_1}, \frac{y}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y))
        (t_2 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_3 (fma z (- t a) (* x y)))
        (t_4 (/ z t_3)))
   (if (<= t_2 -5e+280)
     (-
      (- (/ x (+ z -1.0)))
      (/
       (fma
        z
        (/ (- t a) (+ z -1.0))
        (/ (* z (* x b)) (* (+ z -1.0) (+ z -1.0))))
       y))
     (if (<= t_2 5e+275)
       (/ 1.0 (- (fma b t_4 (/ y t_3)) (* y t_4)))
       (if (<= t_2 INFINITY)
         (* x (fma z (/ (- t a) (* x t_1)) (/ 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(z, (b - y), y);
	double t_2 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_3 = fma(z, (t - a), (x * y));
	double t_4 = z / t_3;
	double tmp;
	if (t_2 <= -5e+280) {
		tmp = -(x / (z + -1.0)) - (fma(z, ((t - a) / (z + -1.0)), ((z * (x * b)) / ((z + -1.0) * (z + -1.0)))) / y);
	} else if (t_2 <= 5e+275) {
		tmp = 1.0 / (fma(b, t_4, (y / t_3)) - (y * t_4));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = x * fma(z, ((t - a) / (x * t_1)), (y / t_1));
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_3 = fma(z, Float64(t - a), Float64(x * y))
	t_4 = Float64(z / t_3)
	tmp = 0.0
	if (t_2 <= -5e+280)
		tmp = Float64(Float64(-Float64(x / Float64(z + -1.0))) - Float64(fma(z, Float64(Float64(t - a) / Float64(z + -1.0)), Float64(Float64(z * Float64(x * b)) / Float64(Float64(z + -1.0) * Float64(z + -1.0)))) / y));
	elseif (t_2 <= 5e+275)
		tmp = Float64(1.0 / Float64(fma(b, t_4, Float64(y / t_3)) - Float64(y * t_4)));
	elseif (t_2 <= Inf)
		tmp = Float64(x * fma(z, Float64(Float64(t - a) / Float64(x * t_1)), 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[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(t - a), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(z / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+280], N[((-N[(x / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]) - N[(N[(z * N[(N[(t - a), $MachinePrecision] / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(x * b), $MachinePrecision]), $MachinePrecision] / N[(N[(z + -1.0), $MachinePrecision] * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+275], N[(1.0 / N[(N[(b * t$95$4 + N[(y / t$95$3), $MachinePrecision]), $MachinePrecision] - N[(y * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(x * N[(z * N[(N[(t - a), $MachinePrecision] / N[(x * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -5.0000000000000002e280

   1. Initial program 45.0%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 84.0%

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

   if -5.0000000000000002e280 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 5.0000000000000003e275

   1. Initial program 92.3%

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

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 92.1

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 92.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 92.1

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

   7. Applied rewrites 98.5%

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

   if 5.0000000000000003e275 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 34.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 85.4

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

   5. Applied rewrites 85.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 78.0

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

   5. Applied rewrites 78.0%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -5 \cdot 10^{+280}:\\ \;\;\;\;\left(-\frac{x}{z + -1}\right) - \frac{\mathsf{fma}\left(z, \frac{t - a}{z + -1}, \frac{z \cdot \left(x \cdot b\right)}{\left(z + -1\right) \cdot \left(z + -1\right)}\right)}{y}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, \frac{z}{\mathsf{fma}\left(z, t - a, x \cdot y\right)}, \frac{y}{\mathsf{fma}\left(z, t - a, x \cdot y\right)}\right) - y \cdot \frac{z}{\mathsf{fma}\left(z, t - a, x \cdot y\right)}}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, \frac{t - a}{x \cdot \mathsf{fma}\left(z, b - y, y\right)}, \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{\mathsf{fma}\left(z, t, \mathsf{fma}\left(z, -a, x \cdot y\right)\right)}{t\_1}\\ t_3 := \frac{t - a}{b - y}\\ t_4 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_5 := \mathsf{fma}\left(z, b - y, y\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\left(-\frac{x}{z + -1}\right) - \frac{\mathsf{fma}\left(z, \frac{t - a}{z + -1}, \frac{z \cdot \left(x \cdot b\right)}{\left(z + -1\right) \cdot \left(z + -1\right)}\right)}{y}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-277}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+281}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z, \frac{t - a}{x \cdot t\_5}, \frac{y}{t\_5}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (fma z t (fma z (- a) (* x y))) t_1))
        (t_3 (/ (- t a) (- b y)))
        (t_4 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_5 (fma z (- b y) y)))
   (if (<= t_4 (- INFINITY))
     (-
      (- (/ x (+ z -1.0)))
      (/
       (fma
        z
        (/ (- t a) (+ z -1.0))
        (/ (* z (* x b)) (* (+ z -1.0) (+ z -1.0))))
       y))
     (if (<= t_4 -1e-277)
       t_2
       (if (<= t_4 0.0)
         t_3
         (if (<= t_4 2e+281)
           t_2
           (if (<= t_4 INFINITY)
             (* x (fma z (/ (- t a) (* x t_5)) (/ y t_5)))
             t_3)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double t_2 = fma(z, t, fma(z, -a, (x * y))) / t_1;
	double t_3 = (t - a) / (b - y);
	double t_4 = ((x * y) + (z * (t - a))) / t_1;
	double t_5 = fma(z, (b - y), y);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = -(x / (z + -1.0)) - (fma(z, ((t - a) / (z + -1.0)), ((z * (x * b)) / ((z + -1.0) * (z + -1.0)))) / y);
	} else if (t_4 <= -1e-277) {
		tmp = t_2;
	} else if (t_4 <= 0.0) {
		tmp = t_3;
	} else if (t_4 <= 2e+281) {
		tmp = t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = x * fma(z, ((t - a) / (x * t_5)), (y / t_5));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = Float64(fma(z, t, fma(z, Float64(-a), Float64(x * y))) / t_1)
	t_3 = Float64(Float64(t - a) / Float64(b - y))
	t_4 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_5 = fma(z, Float64(b - y), y)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(-Float64(x / Float64(z + -1.0))) - Float64(fma(z, Float64(Float64(t - a) / Float64(z + -1.0)), Float64(Float64(z * Float64(x * b)) / Float64(Float64(z + -1.0) * Float64(z + -1.0)))) / y));
	elseif (t_4 <= -1e-277)
		tmp = t_2;
	elseif (t_4 <= 0.0)
		tmp = t_3;
	elseif (t_4 <= 2e+281)
		tmp = t_2;
	elseif (t_4 <= Inf)
		tmp = Float64(x * fma(z, Float64(Float64(t - a) / Float64(x * t_5)), Float64(y / t_5)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t + N[(z * (-a) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[((-N[(x / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]) - N[(N[(z * N[(N[(t - a), $MachinePrecision] / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(z * N[(x * b), $MachinePrecision]), $MachinePrecision] / N[(N[(z + -1.0), $MachinePrecision] * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -1e-277], t$95$2, If[LessEqual[t$95$4, 0.0], t$95$3, If[LessEqual[t$95$4, 2e+281], t$95$2, If[LessEqual[t$95$4, Infinity], N[(x * N[(z * N[(N[(t - a), $MachinePrecision] / N[(x * t$95$5), $MachinePrecision]), $MachinePrecision] + N[(y / t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 41.3%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 83.0%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.99999999999999969e-278 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 2.0000000000000001e281

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      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 99.5

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

   4. Applied rewrites 99.5%

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

   if -9.99999999999999969e-278 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 10.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 80.5

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

   5. Applied rewrites 80.5%

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

   if 2.0000000000000001e281 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 30.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 84.7

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

   5. Applied rewrites 84.7%

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

Alternative 3: 90.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{\mathsf{fma}\left(z, t, \mathsf{fma}\left(z, -a, x \cdot y\right)\right)}{t\_1}\\ t_3 := \mathsf{fma}\left(z, b - y, y\right)\\ t_4 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_5 := x \cdot \mathsf{fma}\left(z, \frac{t - a}{x \cdot t\_3}, \frac{y}{t\_3}\right)\\ t_6 := \frac{t - a}{b - y}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-277}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+281}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_6\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (fma z t (fma z (- a) (* x y))) t_1))
        (t_3 (fma z (- b y) y))
        (t_4 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_5 (* x (fma z (/ (- t a) (* x t_3)) (/ y t_3))))
        (t_6 (/ (- t a) (- b y))))
   (if (<= t_4 (- INFINITY))
     t_5
     (if (<= t_4 -1e-277)
       t_2
       (if (<= t_4 0.0)
         t_6
         (if (<= t_4 2e+281) t_2 (if (<= t_4 INFINITY) t_5 t_6)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double t_2 = fma(z, t, fma(z, -a, (x * y))) / t_1;
	double t_3 = fma(z, (b - y), y);
	double t_4 = ((x * y) + (z * (t - a))) / t_1;
	double t_5 = x * fma(z, ((t - a) / (x * t_3)), (y / t_3));
	double t_6 = (t - a) / (b - y);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_4 <= -1e-277) {
		tmp = t_2;
	} else if (t_4 <= 0.0) {
		tmp = t_6;
	} else if (t_4 <= 2e+281) {
		tmp = t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_5;
	} else {
		tmp = t_6;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = Float64(fma(z, t, fma(z, Float64(-a), Float64(x * y))) / t_1)
	t_3 = fma(z, Float64(b - y), y)
	t_4 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_5 = Float64(x * fma(z, Float64(Float64(t - a) / Float64(x * t_3)), Float64(y / t_3)))
	t_6 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_4 <= -1e-277)
		tmp = t_2;
	elseif (t_4 <= 0.0)
		tmp = t_6;
	elseif (t_4 <= 2e+281)
		tmp = t_2;
	elseif (t_4 <= Inf)
		tmp = t_5;
	else
		tmp = t_6;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t + N[(z * (-a) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(x * N[(z * N[(N[(t - a), $MachinePrecision] / N[(x * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(y / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$5, If[LessEqual[t$95$4, -1e-277], t$95$2, If[LessEqual[t$95$4, 0.0], t$95$6, If[LessEqual[t$95$4, 2e+281], t$95$2, If[LessEqual[t$95$4, Infinity], t$95$5, t$95$6]]]]]]]]]]]

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)))) < -inf.0 or 2.0000000000000001e281 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 37.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 81.6

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

   5. Applied rewrites 81.6%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.99999999999999969e-278 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 2.0000000000000001e281

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      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 99.5

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

   4. Applied rewrites 99.5%

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

   if -9.99999999999999969e-278 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 10.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 80.5

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

   5. Applied rewrites 80.5%

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

Alternative 4: 83.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, \mathsf{fma}\left(z, -a, x \cdot y\right)\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4e+88)
     t_1
     (if (<= z 6.2e+19)
       (/ (fma z t (fma z (- a) (* x y))) (+ y (* z (- b 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 <= -4e+88) {
		tmp = t_1;
	} else if (z <= 6.2e+19) {
		tmp = fma(z, t, fma(z, -a, (x * y))) / (y + (z * (b - 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 <= -4e+88)
		tmp = t_1;
	elseif (z <= 6.2e+19)
		tmp = Float64(fma(z, t, fma(z, Float64(-a), Float64(x * y))) / Float64(y + Float64(z * Float64(b - 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, -4e+88], t$95$1, If[LessEqual[z, 6.2e+19], N[(N[(z * t + N[(z * (-a) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.99999999999999984e88 or 6.2e19 < z

   1. Initial program 45.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 85.0

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

   5. Applied rewrites 85.0%

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

   if -3.99999999999999984e88 < z < 6.2e19

   1. Initial program 88.4%

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

      1. lift-+.f64 N/A

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

      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 88.4

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

   4. Applied rewrites 88.4%

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

Alternative 5: 69.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -6.7 \cdot 10^{-41}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-279}:\\ \;\;\;\;x \cdot \frac{y}{t\_1}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t}{y}, x\right)\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -6.7e-41)
     t_2
     (if (<= z 2.1e-279)
       (* x (/ y t_1))
       (if (<= z 4.6e-113)
         (fma z (/ t y) x)
         (if (<= z 5.2e+14) (/ (* z (- t a)) t_1) t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -6.7e-41) {
		tmp = t_2;
	} else if (z <= 2.1e-279) {
		tmp = x * (y / t_1);
	} else if (z <= 4.6e-113) {
		tmp = fma(z, (t / y), x);
	} else if (z <= 5.2e+14) {
		tmp = (z * (t - a)) / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -6.7e-41)
		tmp = t_2;
	elseif (z <= 2.1e-279)
		tmp = Float64(x * Float64(y / t_1));
	elseif (z <= 4.6e-113)
		tmp = fma(z, Float64(t / y), x);
	elseif (z <= 5.2e+14)
		tmp = Float64(Float64(z * Float64(t - a)) / t_1);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.7e-41], t$95$2, If[LessEqual[z, 2.1e-279], N[(x * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.6e-113], N[(z * N[(t / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 5.2e+14], N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.69999999999999957e-41 or 5.2e14 < z

   1. Initial program 55.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if -6.69999999999999957e-41 < z < 2.10000000000000006e-279

   1. Initial program 91.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if 2.10000000000000006e-279 < z < 4.60000000000000016e-113

   1. Initial program 81.1%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 81.1

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

   4. Applied rewrites 81.1%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 68.1

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

   7. Applied rewrites 68.1%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 82.2%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 82.0%

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

   if 4.60000000000000016e-113 < z < 5.2e14

   1. Initial program 87.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 70.9

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

   5. Applied rewrites 70.9%

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

Alternative 6: 83.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4e+88)
     t_1
     (if (<= z 6.2e+19)
       (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b 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 <= -4e+88) {
		tmp = t_1;
	} else if (z <= 6.2e+19) {
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - 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 <= (-4d+88)) then
        tmp = t_1
    else if (z <= 6.2d+19) then
        tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - 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 <= -4e+88) {
		tmp = t_1;
	} else if (z <= 6.2e+19) {
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - 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 <= -4e+88:
		tmp = t_1
	elif z <= 6.2e+19:
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - 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 <= -4e+88)
		tmp = t_1;
	elseif (z <= 6.2e+19)
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - 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 <= -4e+88)
		tmp = t_1;
	elseif (z <= 6.2e+19)
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - 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, -4e+88], t$95$1, If[LessEqual[z, 6.2e+19], 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], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.99999999999999984e88 or 6.2e19 < z

   1. Initial program 45.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 85.0

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

   5. Applied rewrites 85.0%

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

   if -3.99999999999999984e88 < z < 6.2e19

   1. Initial program 88.4%

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

Alternative 7: 71.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4.9 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+19}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4.9e+73)
     t_1
     (if (<= z 1.65e+19) (/ (fma z t (* x y)) (+ y (* z (- b 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 <= -4.9e+73) {
		tmp = t_1;
	} else if (z <= 1.65e+19) {
		tmp = fma(z, t, (x * y)) / (y + (z * (b - 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 <= -4.9e+73)
		tmp = t_1;
	elseif (z <= 1.65e+19)
		tmp = Float64(fma(z, t, Float64(x * y)) / Float64(y + Float64(z * Float64(b - 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, -4.9e+73], t$95$1, If[LessEqual[z, 1.65e+19], N[(N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8999999999999999e73 or 1.65e19 < z

   1. Initial program 47.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 84.8

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

   5. Applied rewrites 84.8%

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

   if -4.8999999999999999e73 < z < 1.65e19

   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

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 71.3

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

   5. Applied rewrites 71.3%

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

Alternative 8: 71.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4.9 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+19}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot t\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4.9e+73)
     t_1
     (if (<= z 1.65e+19) (/ (fma x y (* z t)) (fma z (- b y) y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -4.9e+73) {
		tmp = t_1;
	} else if (z <= 1.65e+19) {
		tmp = fma(x, y, (z * t)) / fma(z, (b - y), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -4.9e+73)
		tmp = t_1;
	elseif (z <= 1.65e+19)
		tmp = Float64(fma(x, y, Float64(z * t)) / fma(z, Float64(b - y), y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.9e+73], t$95$1, If[LessEqual[z, 1.65e+19], N[(N[(x * y + N[(z * t), $MachinePrecision]), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8999999999999999e73 or 1.65e19 < z

   1. Initial program 47.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 84.8

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

   5. Applied rewrites 84.8%

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

   if -4.8999999999999999e73 < z < 1.65e19

   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. 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 88.7

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 71.3

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

   7. Applied rewrites 71.3%

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

Alternative 9: 71.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4.9 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+19}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4.9e+73)
     t_1
     (if (<= z 1.65e+19) (/ (fma z t (* x y)) (fma z (- b y) y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -4.9e+73) {
		tmp = t_1;
	} else if (z <= 1.65e+19) {
		tmp = fma(z, t, (x * y)) / fma(z, (b - y), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -4.9e+73)
		tmp = t_1;
	elseif (z <= 1.65e+19)
		tmp = Float64(fma(z, t, Float64(x * y)) / fma(z, Float64(b - y), y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.9e+73], t$95$1, If[LessEqual[z, 1.65e+19], N[(N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8999999999999999e73 or 1.65e19 < z

   1. Initial program 47.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 84.8

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

   5. Applied rewrites 84.8%

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

   if -4.8999999999999999e73 < z < 1.65e19

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 71.3

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

   5. Applied rewrites 71.3%

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

Alternative 10: 68.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -6.7 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-279}:\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -6.7e-41)
     t_1
     (if (<= z 2.1e-279)
       (* x (/ y (fma z (- b y) y)))
       (if (<= z 5e-27) (fma z (/ t 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 <= -6.7e-41) {
		tmp = t_1;
	} else if (z <= 2.1e-279) {
		tmp = x * (y / fma(z, (b - y), y));
	} else if (z <= 5e-27) {
		tmp = fma(z, (t / 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 <= -6.7e-41)
		tmp = t_1;
	elseif (z <= 2.1e-279)
		tmp = Float64(x * Float64(y / fma(z, Float64(b - y), y)));
	elseif (z <= 5e-27)
		tmp = fma(z, Float64(t / 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, -6.7e-41], t$95$1, If[LessEqual[z, 2.1e-279], N[(x * N[(y / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5e-27], N[(z * N[(t / y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.69999999999999957e-41 or 5.0000000000000002e-27 < z

   1. Initial program 57.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if -6.69999999999999957e-41 < z < 2.10000000000000006e-279

   1. Initial program 91.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if 2.10000000000000006e-279 < z < 5.0000000000000002e-27

   1. Initial program 84.4%

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

      1. lift-+.f64 N/A

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

      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 84.4

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

   4. Applied rewrites 84.4%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 66.7

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

   7. Applied rewrites 66.7%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 73.2%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 73.3%

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

Alternative 11: 55.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-71}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -4.8e-18)
     t_1
     (if (<= y -2.7e-71)
       (/ t (- b y))
       (if (<= y 3.5e+26) (/ (- 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 <= -4.8e-18) {
		tmp = t_1;
	} else if (y <= -2.7e-71) {
		tmp = t / (b - y);
	} else if (y <= 3.5e+26) {
		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 <= (-4.8d-18)) then
        tmp = t_1
    else if (y <= (-2.7d-71)) then
        tmp = t / (b - y)
    else if (y <= 3.5d+26) 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 <= -4.8e-18) {
		tmp = t_1;
	} else if (y <= -2.7e-71) {
		tmp = t / (b - y);
	} else if (y <= 3.5e+26) {
		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 <= -4.8e-18:
		tmp = t_1
	elif y <= -2.7e-71:
		tmp = t / (b - y)
	elif y <= 3.5e+26:
		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 <= -4.8e-18)
		tmp = t_1;
	elseif (y <= -2.7e-71)
		tmp = Float64(t / Float64(b - y));
	elseif (y <= 3.5e+26)
		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 <= -4.8e-18)
		tmp = t_1;
	elseif (y <= -2.7e-71)
		tmp = t / (b - y);
	elseif (y <= 3.5e+26)
		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, -4.8e-18], t$95$1, If[LessEqual[y, -2.7e-71], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e+26], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.79999999999999988e-18 or 3.4999999999999999e26 < y

   1. Initial program 57.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 58.5

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

   5. Applied rewrites 58.5%

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

   if -4.79999999999999988e-18 < y < -2.7000000000000001e-71

   1. Initial program 67.7%

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

      1. lift-+.f64 N/A

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

      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 67.8

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

   4. Applied rewrites 67.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 54.3

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 61.4%

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

   if -2.7000000000000001e-71 < y < 3.4999999999999999e26

   1. Initial program 84.3%

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

   3. Taylor expanded in y around 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 60.3

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

   5. Applied rewrites 60.3%

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

Alternative 12: 68.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2.25 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -2.25e-73) t_1 (if (<= z 5e-27) (fma z (/ t 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 <= -2.25e-73) {
		tmp = t_1;
	} else if (z <= 5e-27) {
		tmp = fma(z, (t / 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 <= -2.25e-73)
		tmp = t_1;
	elseif (z <= 5e-27)
		tmp = fma(z, Float64(t / 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, -2.25e-73], t$95$1, If[LessEqual[z, 5e-27], N[(z * N[(t / y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.25e-73 or 5.0000000000000002e-27 < z

   1. Initial program 59.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 74.9

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

   5. Applied rewrites 74.9%

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

   if -2.25e-73 < z < 5.0000000000000002e-27

   1. Initial program 88.5%

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

      1. lift-+.f64 N/A

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

      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 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 74.3

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

   7. Applied rewrites 74.3%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 67.9%

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

   11. Taylor expanded in t around inf

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

   13. Applied rewrites 69.9%

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

Alternative 13: 64.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -2.8e-93) t_1 (if (<= z 5.8e-53) (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 <= -2.8e-93) {
		tmp = t_1;
	} else if (z <= 5.8e-53) {
		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 <= -2.8e-93)
		tmp = t_1;
	elseif (z <= 5.8e-53)
		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, -2.8e-93], t$95$1, If[LessEqual[z, 5.8e-53], N[(x * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.79999999999999998e-93 or 5.7999999999999996e-53 < z

   1. Initial program 61.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if -2.79999999999999998e-93 < z < 5.7999999999999996e-53

   1. Initial program 87.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 58.9

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 58.9%

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

Alternative 14: 55.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -6.3 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -6.3e-30) t_1 (if (<= y 3.5e+26) (/ (- 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 <= -6.3e-30) {
		tmp = t_1;
	} else if (y <= 3.5e+26) {
		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 <= (-6.3d-30)) then
        tmp = t_1
    else if (y <= 3.5d+26) 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 <= -6.3e-30) {
		tmp = t_1;
	} else if (y <= 3.5e+26) {
		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 <= -6.3e-30:
		tmp = t_1
	elif y <= 3.5e+26:
		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 <= -6.3e-30)
		tmp = t_1;
	elseif (y <= 3.5e+26)
		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 <= -6.3e-30)
		tmp = t_1;
	elseif (y <= 3.5e+26)
		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, -6.3e-30], t$95$1, If[LessEqual[y, 3.5e+26], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.29999999999999982e-30 or 3.4999999999999999e26 < y

   1. Initial program 58.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 58.0

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

   5. Applied rewrites 58.0%

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

   if -6.29999999999999982e-30 < y < 3.4999999999999999e26

   1. Initial program 82.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 57.3

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

   5. Applied rewrites 57.3%

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

Alternative 15: 43.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -5.6 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-53}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -5.6e-39) t_1 (if (<= y 1.15e-53) (/ t b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -5.6e-39) {
		tmp = t_1;
	} else if (y <= 1.15e-53) {
		tmp = t / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (1.0d0 - z)
    if (y <= (-5.6d-39)) then
        tmp = t_1
    else if (y <= 1.15d-53) then
        tmp = t / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -5.6e-39) {
		tmp = t_1;
	} else if (y <= 1.15e-53) {
		tmp = t / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -5.6e-39:
		tmp = t_1
	elif y <= 1.15e-53:
		tmp = t / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -5.6e-39)
		tmp = t_1;
	elseif (y <= 1.15e-53)
		tmp = Float64(t / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -5.6e-39)
		tmp = t_1;
	elseif (y <= 1.15e-53)
		tmp = t / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.6e-39], t$95$1, If[LessEqual[y, 1.15e-53], N[(t / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.6000000000000003e-39 or 1.1500000000000001e-53 < y

   1. Initial program 60.4%

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

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

   if -5.6000000000000003e-39 < y < 1.1500000000000001e-53

   1. Initial program 82.7%

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

      1. lift-+.f64 N/A

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

      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 82.7

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

   4. Applied rewrites 82.7%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 52.6

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

   7. Applied rewrites 52.6%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 35.2%

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

Alternative 16: 37.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{-73}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.25e-73) (/ t b) (if (<= z 5.2e-62) (/ x 1.0) (- (/ a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.25e-73) {
		tmp = t / b;
	} else if (z <= 5.2e-62) {
		tmp = x / 1.0;
	} else {
		tmp = -(a / b);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-2.25d-73)) then
        tmp = t / b
    else if (z <= 5.2d-62) then
        tmp = x / 1.0d0
    else
        tmp = -(a / b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.25e-73) {
		tmp = t / b;
	} else if (z <= 5.2e-62) {
		tmp = x / 1.0;
	} else {
		tmp = -(a / b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -2.25e-73:
		tmp = t / b
	elif z <= 5.2e-62:
		tmp = x / 1.0
	else:
		tmp = -(a / b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2.25e-73)
		tmp = Float64(t / b);
	elseif (z <= 5.2e-62)
		tmp = Float64(x / 1.0);
	else
		tmp = Float64(-Float64(a / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -2.25e-73)
		tmp = t / b;
	elseif (z <= 5.2e-62)
		tmp = x / 1.0;
	else
		tmp = -(a / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2.25e-73], N[(t / b), $MachinePrecision], If[LessEqual[z, 5.2e-62], N[(x / 1.0), $MachinePrecision], (-N[(a / b), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if z < -2.25e-73

   1. Initial program 58.1%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 58.1

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

   4. Applied rewrites 58.1%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 44.5

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

   7. Applied rewrites 44.5%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 36.4%

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

   if -2.25e-73 < z < 5.1999999999999999e-62

   1. Initial program 88.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 58.7%

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

   if 5.1999999999999999e-62 < z

   1. Initial program 62.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. neg-mul-1 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 39.5

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

   5. Applied rewrites 39.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 25.0%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{-73}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;-\frac{a}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 36.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.25 \cdot 10^{-73}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 3.35 \cdot 10^{-62}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.25e-73) (/ t b) (if (<= z 3.35e-62) (/ x 1.0) (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.25e-73) {
		tmp = t / b;
	} else if (z <= 3.35e-62) {
		tmp = x / 1.0;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-2.25d-73)) then
        tmp = t / b
    else if (z <= 3.35d-62) then
        tmp = x / 1.0d0
    else
        tmp = t / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.25e-73) {
		tmp = t / b;
	} else if (z <= 3.35e-62) {
		tmp = x / 1.0;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -2.25e-73:
		tmp = t / b
	elif z <= 3.35e-62:
		tmp = x / 1.0
	else:
		tmp = t / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2.25e-73)
		tmp = Float64(t / b);
	elseif (z <= 3.35e-62)
		tmp = Float64(x / 1.0);
	else
		tmp = Float64(t / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -2.25e-73)
		tmp = t / b;
	elseif (z <= 3.35e-62)
		tmp = x / 1.0;
	else
		tmp = t / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2.25e-73], N[(t / b), $MachinePrecision], If[LessEqual[z, 3.35e-62], N[(x / 1.0), $MachinePrecision], N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.25e-73 or 3.34999999999999996e-62 < z

   1. Initial program 60.5%

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

      1. lift-+.f64 N/A

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

      2. +-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 60.6

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

   4. Applied rewrites 60.6%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 41.1

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

   7. Applied rewrites 41.1%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 29.2%

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

   if -2.25e-73 < z < 3.34999999999999996e-62

   1. Initial program 88.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 58.7%

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

Alternative 18: 25.6% 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 70.8%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
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 34.5

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

5. Applied rewrites 34.5%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 26.2%

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

Alternative 19: 3.9% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
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 34.5

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

5. Applied rewrites 34.5%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 26.2%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 4.7%

    \[\leadsto z \cdot x \]

12. Final simplification 4.7%

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

Developer Target 1: 73.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(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)))))