Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.3% → 94.2%

Time: 16.5s

Alternatives: 20

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% 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: 94.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, t - a, x \cdot y\right)\\ t_2 := \frac{z}{t\_1}\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_4 := \mathsf{fma}\left(z, b - y, y\right)\\ t_5 := \frac{t - a}{t\_4}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, t\_5, \frac{x}{1 - z}\right)\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, t\_2, \frac{y}{t\_1}\right) - y \cdot t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, t\_5, x \cdot \frac{y}{t\_4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- t a) (* x y)))
        (t_2 (/ z t_1))
        (t_3 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_4 (fma z (- b y) y))
        (t_5 (/ (- t a) t_4)))
   (if (<= t_3 (- INFINITY))
     (fma z t_5 (/ x (- 1.0 z)))
     (if (<= t_3 5e+275)
       (/ 1.0 (- (fma b t_2 (/ y t_1)) (* y t_2)))
       (if (<= t_3 INFINITY)
         (fma z t_5 (* x (/ y t_4)))
         (/ (- t a) (- b y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (t - a), (x * y));
	double t_2 = z / t_1;
	double t_3 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_4 = fma(z, (b - y), y);
	double t_5 = (t - a) / t_4;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma(z, t_5, (x / (1.0 - z)));
	} else if (t_3 <= 5e+275) {
		tmp = 1.0 / (fma(b, t_2, (y / t_1)) - (y * t_2));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = fma(z, t_5, (x * (y / t_4)));
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(t - a), Float64(x * y))
	t_2 = Float64(z / t_1)
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_4 = fma(z, Float64(b - y), y)
	t_5 = Float64(Float64(t - a) / t_4)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = fma(z, t_5, Float64(x / Float64(1.0 - z)));
	elseif (t_3 <= 5e+275)
		tmp = Float64(1.0 / Float64(fma(b, t_2, Float64(y / t_1)) - Float64(y * t_2)));
	elseif (t_3 <= Inf)
		tmp = fma(z, t_5, Float64(x * Float64(y / t_4)));
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(t - a), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z / t$95$1), $MachinePrecision]}, Block[{t$95$3 = 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$4 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t - a), $MachinePrecision] / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(z * t$95$5 + N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5e+275], N[(1.0 / N[(N[(b * t$95$2 + N[(y / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(y * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(z * t$95$5 + N[(x * N[(y / t$95$4), $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)))) < -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 x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower--.f64 99.8

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

   8. Applied rewrites 99.8%

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

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

   1. Initial program 92.4%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 92.2

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 92.2

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 92.2

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

   4. Applied rewrites 92.2%

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

   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. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

   7. Applied rewrites 98.5%

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

   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 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   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 96.2%

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

Alternative 2: 93.5% accurate, 0.1× speedup

Math

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

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 (/ (- t a) t_3))
        (t_5 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_6 (/ (- t a) (- b y))))
   (if (<= t_5 (- INFINITY))
     (fma z t_4 (/ x (- 1.0 z)))
     (if (<= t_5 -1e-277)
       t_2
       (if (<= t_5 0.0)
         t_6
         (if (<= t_5 2e+281)
           t_2
           (if (<= t_5 INFINITY) (fma z t_4 (* x (/ y t_3))) 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 = (t - a) / t_3;
	double t_5 = ((x * y) + (z * (t - a))) / t_1;
	double t_6 = (t - a) / (b - y);
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = fma(z, t_4, (x / (1.0 - z)));
	} else if (t_5 <= -1e-277) {
		tmp = t_2;
	} else if (t_5 <= 0.0) {
		tmp = t_6;
	} else if (t_5 <= 2e+281) {
		tmp = t_2;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = fma(z, t_4, (x * (y / t_3)));
	} 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(t - a) / t_3)
	t_5 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_6 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = fma(z, t_4, Float64(x / Float64(1.0 - z)));
	elseif (t_5 <= -1e-277)
		tmp = t_2;
	elseif (t_5 <= 0.0)
		tmp = t_6;
	elseif (t_5 <= 2e+281)
		tmp = t_2;
	elseif (t_5 <= Inf)
		tmp = fma(z, t_4, Float64(x * Float64(y / t_3)));
	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[(t - a), $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], N[(z * t$95$4 + N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, -1e-277], t$95$2, If[LessEqual[t$95$5, 0.0], t$95$6, If[LessEqual[t$95$5, 2e+281], t$95$2, If[LessEqual[t$95$5, Infinity], N[(z * t$95$4 + N[(x * N[(y / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$6]]]]]]]]]]]

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 x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower--.f64 99.8

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

   8. Applied rewrites 99.8%

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

   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. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+l+ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. 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 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

Alternative 3: 93.1% accurate, 0.1× speedup

Math

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

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

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 = ((x * y) + (z * (t - a))) / t_1;
	double t_4 = fma(z, ((t - a) / fma(z, (b - y), y)), (x / (1.0 - z)));
	double t_5 = (t - a) / (b - y);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -1e-277) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = t_5;
	} else if (t_3 <= 2e+281) {
		tmp = t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = t_5;
	}
	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(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_4 = fma(z, Float64(Float64(t - a) / fma(z, Float64(b - y), y)), Float64(x / Float64(1.0 - z)))
	t_5 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -1e-277)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = t_5;
	elseif (t_3 <= 2e+281)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = t_5;
	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[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(N[(t - a), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -1e-277], t$95$2, If[LessEqual[t$95$3, 0.0], t$95$5, If[LessEqual[t$95$3, 2e+281], t$95$2, If[LessEqual[t$95$3, Infinity], t$95$4, t$95$5]]]]]]]]]]

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 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower--.f64 96.1

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

   8. Applied rewrites 96.1%

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

   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. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+l+ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. 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: 90.9% accurate, 0.2× speedup

Math

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

FPCore

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

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

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	t_3 = Float64(fma(z, t, fma(z, Float64(-a), Float64(x * y))) / t_1)
	t_4 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_5 = fma(z, Float64(Float64(t - a) / fma(z, Float64(b - y), y)), x)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_4 <= -1e-277)
		tmp = t_3;
	elseif (t_4 <= 0.0)
		tmp = t_2;
	elseif (t_4 <= 2e+281)
		tmp = t_3;
	elseif (t_4 <= Inf)
		tmp = t_5;
	else
		tmp = t_2;
	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[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z * t + N[(z * (-a) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $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[(N[(t - a), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$5, If[LessEqual[t$95$4, -1e-277], t$95$3, If[LessEqual[t$95$4, 0.0], t$95$2, If[LessEqual[t$95$4, 2e+281], t$95$3, If[LessEqual[t$95$4, Infinity], t$95$5, t$95$2]]]]]]]]]]

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 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 85.5%

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

   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. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. sub-neg N/A

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

      8. distribute-lft-in N/A

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

      9. associate-+l+ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. 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. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-277}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, \mathsf{fma}\left(z, -a, x \cdot y\right)\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 2 \cdot 10^{+281}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, \mathsf{fma}\left(z, -a, x \cdot y\right)\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_3 := \mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-277}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+275}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_3 (fma z (/ (- t a) (fma z (- b y) y)) x)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -1e-277)
       t_2
       (if (<= t_2 0.0)
         t_1
         (if (<= t_2 5e+275) t_2 (if (<= t_2 INFINITY) t_3 t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double t_2 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_3 = fma(z, ((t - a) / fma(z, (b - y), y)), x);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -1e-277) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+275) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_3 = fma(z, Float64(Float64(t - a) / fma(z, Float64(b - y), y)), x)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -1e-277)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 5e+275)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(N[(t - a), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -1e-277], t$95$2, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 5e+275], t$95$2, If[LessEqual[t$95$2, Infinity], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 38.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 85.8%

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

   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)))) < 5.0000000000000003e275

   1. Initial program 99.5%

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

   if -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. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-277}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(z, b - y, y\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -5 \cdot 10^{+88}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t\_1}, x, \frac{z \cdot \left(t - a\right)}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -5e+88)
     t_2
     (if (<= z 6.2e+19) (fma (/ y t_1) x (/ (* 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 <= -5e+88) {
		tmp = t_2;
	} else if (z <= 6.2e+19) {
		tmp = fma((y / t_1), x, ((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 <= -5e+88)
		tmp = t_2;
	elseif (z <= 6.2e+19)
		tmp = fma(Float64(y / t_1), x, 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, -5e+88], t$95$2, If[LessEqual[z, 6.2e+19], N[(N[(y / t$95$1), $MachinePrecision] * x + N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.99999999999999997e88 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 -4.99999999999999997e88 < 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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 89.0

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

   5. Applied rewrites 89.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-/.f64 N/A

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

   7. Applied rewrites 97.8%

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

Alternative 7: 77.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4.9 \cdot 10^{+73}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{t\_1}\\ \mathbf{elif}\;z \leq -2.45 \cdot 10^{-209}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, -a, x \cdot y\right)}{t\_1}\\ \mathbf{elif}\;z \leq 265000:\\ \;\;\;\;x + \frac{z \cdot \left(t - a\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -4.9e+73)
     t_2
     (if (<= z -5.8e-33)
       (/ (fma z t (* x y)) t_1)
       (if (<= z -2.45e-209)
         (/ (fma z (- a) (* x y)) t_1)
         (if (<= z 265000.0) (+ x (/ (* 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 <= -4.9e+73) {
		tmp = t_2;
	} else if (z <= -5.8e-33) {
		tmp = fma(z, t, (x * y)) / t_1;
	} else if (z <= -2.45e-209) {
		tmp = fma(z, -a, (x * y)) / t_1;
	} else if (z <= 265000.0) {
		tmp = x + ((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 <= -4.9e+73)
		tmp = t_2;
	elseif (z <= -5.8e-33)
		tmp = Float64(fma(z, t, Float64(x * y)) / t_1);
	elseif (z <= -2.45e-209)
		tmp = Float64(fma(z, Float64(-a), Float64(x * y)) / t_1);
	elseif (z <= 265000.0)
		tmp = Float64(x + 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, -4.9e+73], t$95$2, If[LessEqual[z, -5.8e-33], N[(N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, -2.45e-209], N[(N[(z * (-a) + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 265000.0], N[(x + N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.8999999999999999e73 or 265000 < z

   1. Initial program 48.2%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 82.7

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

   5. Applied rewrites 82.7%

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

   if -4.8999999999999999e73 < z < -5.80000000000000005e-33

   1. Initial program 90.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. *-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 81.7

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

   5. Applied rewrites 81.7%

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

   if -5.80000000000000005e-33 < z < -2.45000000000000018e-209

   1. Initial program 93.8%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 82.5

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

   5. Applied rewrites 82.5%

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

   if -2.45000000000000018e-209 < z < 265000

   1. Initial program 86.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 74.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 87.5

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

   10. Applied rewrites 87.5%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+73}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq -2.45 \cdot 10^{-209}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, -a, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 265000:\\ \;\;\;\;x + \frac{z \cdot \left(t - a\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4.9 \cdot 10^{+73}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -7.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{t\_1}\\ \mathbf{elif}\;z \leq 265000:\\ \;\;\;\;x + \frac{z \cdot \left(t - a\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -4.9e+73)
     t_2
     (if (<= z -7.4e-34)
       (/ (fma z t (* x y)) t_1)
       (if (<= z 265000.0) (+ x (/ (* 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 <= -4.9e+73) {
		tmp = t_2;
	} else if (z <= -7.4e-34) {
		tmp = fma(z, t, (x * y)) / t_1;
	} else if (z <= 265000.0) {
		tmp = x + ((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 <= -4.9e+73)
		tmp = t_2;
	elseif (z <= -7.4e-34)
		tmp = Float64(fma(z, t, Float64(x * y)) / t_1);
	elseif (z <= 265000.0)
		tmp = Float64(x + 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, -4.9e+73], t$95$2, If[LessEqual[z, -7.4e-34], N[(N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 265000.0], N[(x + N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.8999999999999999e73 or 265000 < z

   1. Initial program 48.2%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 82.7

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

   5. Applied rewrites 82.7%

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

   if -4.8999999999999999e73 < z < -7.39999999999999976e-34

   1. Initial program 90.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 78.8

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

   5. Applied rewrites 78.8%

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

   if -7.39999999999999976e-34 < z < 265000

   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 x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 72.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 82.9

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

   10. Applied rewrites 82.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+73}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -7.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 265000:\\ \;\;\;\;x + \frac{z \cdot \left(t - a\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -9.6 \cdot 10^{-34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 265000:\\ \;\;\;\;x + \frac{z \cdot \left(t - a\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -9.6e-34)
     t_1
     (if (<= z 265000.0) (+ x (/ (* z (- t a)) (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 <= -9.6e-34) {
		tmp = t_1;
	} else if (z <= 265000.0) {
		tmp = x + ((z * (t - a)) / 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 <= -9.6e-34)
		tmp = t_1;
	elseif (z <= 265000.0)
		tmp = Float64(x + Float64(Float64(z * Float64(t - a)) / 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, -9.6e-34], t$95$1, If[LessEqual[z, 265000.0], N[(x + N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.59999999999999965e-34 or 265000 < z

   1. Initial program 55.1%

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

   3. Taylor expanded in 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 78.6

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

   5. Applied rewrites 78.6%

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

   if -9.59999999999999965e-34 < z < 265000

   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 x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 72.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 82.9

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

   10. Applied rewrites 82.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{-34}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 265000:\\ \;\;\;\;x + \frac{z \cdot \left(t - a\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-278}:\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -2.9e-43)
     t_1
     (if (<= z 1.2e-278)
       (* x (/ y (fma z (- b y) y)))
       (if (<= z 7.2e-25) (fma z (/ (- t a) 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 <= -2.9e-43) {
		tmp = t_1;
	} else if (z <= 1.2e-278) {
		tmp = x * (y / fma(z, (b - y), y));
	} else if (z <= 7.2e-25) {
		tmp = fma(z, ((t - a) / 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 <= -2.9e-43)
		tmp = t_1;
	elseif (z <= 1.2e-278)
		tmp = Float64(x * Float64(y / fma(z, Float64(b - y), y)));
	elseif (z <= 7.2e-25)
		tmp = fma(z, Float64(Float64(t - a) / 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, -2.9e-43], t$95$1, If[LessEqual[z, 1.2e-278], N[(x * N[(y / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.2e-25], N[(z * N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.9000000000000001e-43 or 7.1999999999999998e-25 < 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 -2.9000000000000001e-43 < z < 1.2e-278

   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 1.2e-278 < z < 7.1999999999999998e-25

   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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 88.3

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 82.3%

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

   9. Taylor expanded in z around 0

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

       1. div-sub N/A

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

       2. associate-/l* N/A

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

       3. +-commutative N/A

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

       4. associate-/l* N/A

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

       5. div-sub N/A

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

       6. lower-fma.f64 N/A

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

       7. div-sub N/A

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

       8. lower-/.f64 N/A

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

       9. lower--.f64 75.4

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

   11. Applied rewrites 75.4%

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

Alternative 11: 55.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{-19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-71}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -5.8e-19)
     t_1
     (if (<= y -3.1e-71)
       (/ t (- b y))
       (if (<= y 3.8e+24) (/ (- t a) b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -5.8e-19) {
		tmp = t_1;
	} else if (y <= -3.1e-71) {
		tmp = t / (b - y);
	} else if (y <= 3.8e+24) {
		tmp = (t - a) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (1.0d0 - z)
    if (y <= (-5.8d-19)) then
        tmp = t_1
    else if (y <= (-3.1d-71)) then
        tmp = t / (b - y)
    else if (y <= 3.8d+24) then
        tmp = (t - a) / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -5.8e-19) {
		tmp = t_1;
	} else if (y <= -3.1e-71) {
		tmp = t / (b - y);
	} else if (y <= 3.8e+24) {
		tmp = (t - a) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -5.8e-19:
		tmp = t_1
	elif y <= -3.1e-71:
		tmp = t / (b - y)
	elif y <= 3.8e+24:
		tmp = (t - a) / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -5.8e-19)
		tmp = t_1;
	elseif (y <= -3.1e-71)
		tmp = Float64(t / Float64(b - y));
	elseif (y <= 3.8e+24)
		tmp = Float64(Float64(t - a) / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -5.8e-19)
		tmp = t_1;
	elseif (y <= -3.1e-71)
		tmp = t / (b - y);
	elseif (y <= 3.8e+24)
		tmp = (t - a) / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.8e-19], t$95$1, If[LessEqual[y, -3.1e-71], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e+24], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.8e-19 or 3.80000000000000015e24 < 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 -5.8e-19 < y < -3.10000000000000002e-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. 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 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 61.4

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

   8. Applied rewrites 61.4%

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

   if -3.10000000000000002e-71 < y < 3.80000000000000015e24

   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: 71.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2.25 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 7.2e-25) (fma z (/ (- t a) 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 <= 7.2e-25) {
		tmp = fma(z, ((t - a) / 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 <= 7.2e-25)
		tmp = fma(z, Float64(Float64(t - a) / 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, 7.2e-25], N[(z * N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.25e-73 or 7.1999999999999998e-25 < 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 < 7.1999999999999998e-25

   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. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower--.f64 84.8

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 74.4%

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

   9. Taylor expanded in z around 0

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

       1. div-sub N/A

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

       2. associate-/l* N/A

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

       3. +-commutative N/A

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

       4. associate-/l* N/A

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

       5. div-sub N/A

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

       6. lower-fma.f64 N/A

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

       7. div-sub N/A

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

       8. lower-/.f64 N/A

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

       9. lower--.f64 70.0

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

   11. Applied rewrites 70.0%

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

Alternative 13: 64.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -5 \cdot 10^{-94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -5e-94) t_1 (if (<= z 6.4e-52) 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 <= -5e-94) {
		tmp = t_1;
	} else if (z <= 6.4e-52) {
		tmp = x;
	} 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 <= (-5d-94)) then
        tmp = t_1
    else if (z <= 6.4d-52) then
        tmp = x
    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 <= -5e-94) {
		tmp = t_1;
	} else if (z <= 6.4e-52) {
		tmp = x;
	} 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 <= -5e-94:
		tmp = t_1
	elif z <= 6.4e-52:
		tmp = 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 <= -5e-94)
		tmp = t_1;
	elseif (z <= 6.4e-52)
		tmp = x;
	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 <= -5e-94)
		tmp = t_1;
	elseif (z <= 6.4e-52)
		tmp = x;
	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, -5e-94], t$95$1, If[LessEqual[z, 6.4e-52], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.9999999999999995e-94 or 6.4000000000000002e-52 < 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 -4.9999999999999995e-94 < z < 6.4000000000000002e-52

   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. 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. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 87.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 87.5

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 87.5

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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 58.6

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

   7. Applied rewrites 58.6%

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

      1. remove-double-div 58.9

         \[\leadsto \color{blue}{x} \]

   9. Applied rewrites 58.9%

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

Alternative 14: 45.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-41}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{y - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.25e-41)
   (/ t (- b y))
   (if (<= z 6e+18) (/ x (- 1.0 z)) (/ a (- y b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.25e-41) {
		tmp = t / (b - y);
	} else if (z <= 6e+18) {
		tmp = x / (1.0 - z);
	} else {
		tmp = a / (y - 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 <= (-1.25d-41)) then
        tmp = t / (b - y)
    else if (z <= 6d+18) then
        tmp = x / (1.0d0 - z)
    else
        tmp = a / (y - 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 <= -1.25e-41) {
		tmp = t / (b - y);
	} else if (z <= 6e+18) {
		tmp = x / (1.0 - z);
	} else {
		tmp = a / (y - b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -1.25e-41:
		tmp = t / (b - y)
	elif z <= 6e+18:
		tmp = x / (1.0 - z)
	else:
		tmp = a / (y - b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.25e-41)
		tmp = Float64(t / Float64(b - y));
	elseif (z <= 6e+18)
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = Float64(a / Float64(y - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -1.25e-41)
		tmp = t / (b - y);
	elseif (z <= 6e+18)
		tmp = x / (1.0 - z);
	else
		tmp = a / (y - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.25e-41], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6e+18], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(a / N[(y - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.2499999999999999e-41

   1. Initial program 54.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 43.6

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

   5. Applied rewrites 43.6%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 49.0

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

   8. Applied rewrites 49.0%

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

   if -1.2499999999999999e-41 < z < 6e18

   1. Initial program 88.2%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 51.4

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

   5. Applied rewrites 51.4%

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

   if 6e18 < z

   1. Initial program 54.9%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 30.7

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

   5. Applied rewrites 30.7%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 48.6

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

   8. Applied rewrites 48.6%

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

Alternative 15: 45.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-41}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{elif}\;z \leq 1.58 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{y - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.25e-41) (/ t (- b y)) (if (<= z 1.58e-49) x (/ a (- y b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.25e-41) {
		tmp = t / (b - y);
	} else if (z <= 1.58e-49) {
		tmp = x;
	} else {
		tmp = a / (y - 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 <= (-1.25d-41)) then
        tmp = t / (b - y)
    else if (z <= 1.58d-49) then
        tmp = x
    else
        tmp = a / (y - 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 <= -1.25e-41) {
		tmp = t / (b - y);
	} else if (z <= 1.58e-49) {
		tmp = x;
	} else {
		tmp = a / (y - b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -1.25e-41:
		tmp = t / (b - y)
	elif z <= 1.58e-49:
		tmp = x
	else:
		tmp = a / (y - b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.25e-41)
		tmp = Float64(t / Float64(b - y));
	elseif (z <= 1.58e-49)
		tmp = x;
	else
		tmp = Float64(a / Float64(y - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -1.25e-41)
		tmp = t / (b - y);
	elseif (z <= 1.58e-49)
		tmp = x;
	else
		tmp = a / (y - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.25e-41], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.58e-49], x, N[(a / N[(y - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.2499999999999999e-41

   1. Initial program 54.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 43.6

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

   5. Applied rewrites 43.6%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 49.0

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

   8. Applied rewrites 49.0%

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

   if -1.2499999999999999e-41 < z < 1.58e-49

   1. Initial program 88.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 88.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 88.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 88.0

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 54.2

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

   7. Applied rewrites 54.2%

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

      1. remove-double-div 54.4

         \[\leadsto \color{blue}{x} \]

   9. Applied rewrites 54.4%

      \[\leadsto \color{blue}{x} \]

   if 1.58e-49 < z

   1. Initial program 61.9%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 30.3

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

   5. Applied rewrites 30.3%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 44.1

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

   8. Applied rewrites 44.1%

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

Alternative 16: 40.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-73}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 1.58 \cdot 10^{-49}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{y - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.05e-73) (/ t b) (if (<= z 1.58e-49) x (/ a (- y b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.05e-73) {
		tmp = t / b;
	} else if (z <= 1.58e-49) {
		tmp = x;
	} else {
		tmp = a / (y - 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 <= (-1.05d-73)) then
        tmp = t / b
    else if (z <= 1.58d-49) then
        tmp = x
    else
        tmp = a / (y - 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 <= -1.05e-73) {
		tmp = t / b;
	} else if (z <= 1.58e-49) {
		tmp = x;
	} else {
		tmp = a / (y - b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -1.05e-73:
		tmp = t / b
	elif z <= 1.58e-49:
		tmp = x
	else:
		tmp = a / (y - b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.05e-73)
		tmp = Float64(t / b);
	elseif (z <= 1.58e-49)
		tmp = x;
	else
		tmp = Float64(a / Float64(y - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -1.05e-73)
		tmp = t / b;
	elseif (z <= 1.58e-49)
		tmp = x;
	else
		tmp = a / (y - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.05e-73], N[(t / b), $MachinePrecision], If[LessEqual[z, 1.58e-49], x, N[(a / N[(y - b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.0499999999999999e-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. 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 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 36.4

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

   8. Applied rewrites 36.4%

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

   if -1.0499999999999999e-73 < z < 1.58e-49

   1. Initial program 88.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 88.1

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 88.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 88.0

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 57.4

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

   7. Applied rewrites 57.4%

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

      1. remove-double-div 57.6

         \[\leadsto \color{blue}{x} \]

   9. Applied rewrites 57.6%

      \[\leadsto \color{blue}{x} \]

   if 1.58e-49 < z

   1. Initial program 61.9%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 30.3

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

   5. Applied rewrites 30.3%

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

   6. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 44.1

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

   8. Applied rewrites 44.1%

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

Alternative 17: 36.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-73}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.05e-73) (/ t b) (if (<= z 3.8e-62) x (/ a (- b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.05e-73) {
		tmp = t / b;
	} else if (z <= 3.8e-62) {
		tmp = x;
	} 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 <= (-1.05d-73)) then
        tmp = t / b
    else if (z <= 3.8d-62) then
        tmp = x
    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 <= -1.05e-73) {
		tmp = t / b;
	} else if (z <= 3.8e-62) {
		tmp = x;
	} else {
		tmp = a / -b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -1.05e-73:
		tmp = t / b
	elif z <= 3.8e-62:
		tmp = x
	else:
		tmp = a / -b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.05e-73)
		tmp = Float64(t / b);
	elseif (z <= 3.8e-62)
		tmp = x;
	else
		tmp = Float64(a / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -1.05e-73)
		tmp = t / b;
	elseif (z <= 3.8e-62)
		tmp = x;
	else
		tmp = a / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.05e-73], N[(t / b), $MachinePrecision], If[LessEqual[z, 3.8e-62], x, N[(a / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.0499999999999999e-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. 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 44.4

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

   5. Applied rewrites 44.4%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 36.4

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

   8. Applied rewrites 36.4%

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

   if -1.0499999999999999e-73 < z < 3.80000000000000006e-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. 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. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 87.8

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 87.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 87.8

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

   4. Applied rewrites 87.8%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 58.5

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

   7. Applied rewrites 58.5%

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

      1. remove-double-div 58.7

         \[\leadsto \color{blue}{x} \]

   9. Applied rewrites 58.7%

      \[\leadsto \color{blue}{x} \]

   if 3.80000000000000006e-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 a around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 30.8

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in b around inf

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

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{b}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{b}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(a\right)}}{b} \]

      4. lower-neg.f64 25.0

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

   8. Applied rewrites 25.0%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-73}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 36.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-73}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.05e-73) (/ t b) (if (<= z 5.2e-62) x (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.05e-73) {
		tmp = t / b;
	} else if (z <= 5.2e-62) {
		tmp = x;
	} 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 <= (-1.05d-73)) then
        tmp = t / b
    else if (z <= 5.2d-62) then
        tmp = x
    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 <= -1.05e-73) {
		tmp = t / b;
	} else if (z <= 5.2e-62) {
		tmp = x;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -1.05e-73:
		tmp = t / b
	elif z <= 5.2e-62:
		tmp = x
	else:
		tmp = t / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.05e-73)
		tmp = Float64(t / b);
	elseif (z <= 5.2e-62)
		tmp = x;
	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 <= -1.05e-73)
		tmp = t / b;
	elseif (z <= 5.2e-62)
		tmp = x;
	else
		tmp = t / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.05e-73], N[(t / b), $MachinePrecision], If[LessEqual[z, 5.2e-62], x, N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0499999999999999e-73 or 5.1999999999999999e-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. 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 41.1

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

   5. Applied rewrites 41.1%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 29.2

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

   8. Applied rewrites 29.2%

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

   if -1.0499999999999999e-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. 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. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 87.8

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 87.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-fma.f64 87.8

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

   4. Applied rewrites 87.8%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 58.5

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

   7. Applied rewrites 58.5%

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

      1. remove-double-div 58.7

         \[\leadsto \color{blue}{x} \]

   9. Applied rewrites 58.7%

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

Alternative 19: 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

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

   1. +-commutative N/A

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

   2. lower-fma.f64 26.2

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

8. Applied rewrites 26.2%

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

Alternative 20: 25.3% accurate, 39.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

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. 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. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. clear-num N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 70.7

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 70.7

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lower-fma.f64 70.7

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

4. Applied rewrites 70.7%

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

5. Taylor expanded in z around 0

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

   1. lower-/.f64 24.7

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

7. Applied rewrites 24.7%

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

   1. remove-double-div 24.8

      \[\leadsto \color{blue}{x} \]

9. Applied rewrites 24.8%

   \[\leadsto \color{blue}{x} \]
10. 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)))))