Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.4% → 94.1%

Time: 15.6s

Alternatives: 20

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.4% 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: 94.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{y}{t\_1}\\ t_3 := \mathsf{fma}\left(x, t\_2, \frac{t - a}{b - y}\right)\\ \mathbf{if}\;z \leq -960:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq 12200000000000:\\ \;\;\;\;\mathsf{fma}\left(x, t\_2, \frac{z \cdot \left(t - a\right)}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (- b y) y))
        (t_2 (/ y t_1))
        (t_3 (fma x t_2 (/ (- t a) (- b y)))))
   (if (<= z -960.0)
     t_3
     (if (<= z 12200000000000.0) (fma x t_2 (/ (* z (- t a)) t_1)) t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, (b - y), y);
	double t_2 = y / t_1;
	double t_3 = fma(x, t_2, ((t - a) / (b - y)));
	double tmp;
	if (z <= -960.0) {
		tmp = t_3;
	} else if (z <= 12200000000000.0) {
		tmp = fma(x, t_2, ((z * (t - a)) / t_1));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, Float64(b - y), y)
	t_2 = Float64(y / t_1)
	t_3 = fma(x, t_2, Float64(Float64(t - a) / Float64(b - y)))
	tmp = 0.0
	if (z <= -960.0)
		tmp = t_3;
	elseif (z <= 12200000000000.0)
		tmp = fma(x, t_2, Float64(Float64(z * Float64(t - a)) / t_1));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(y / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(x * t$95$2 + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -960.0], t$95$3, If[LessEqual[z, 12200000000000.0], N[(x * t$95$2 + N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if z < -960 or 1.22e13 < z

   1. Initial program 36.3%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 40.8

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

   5. Applied rewrites 40.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 88.9%

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

   if -960 < z < 1.22e13

   1. Initial program 86.8%

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

   3. Taylor expanded in x around 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. associate-/l* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 2: 91.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \mathsf{fma}\left(x, \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, \frac{t - a}{b - y}\right)\\ t_3 := \frac{z \cdot \left(t - a\right) + x \cdot y}{t\_1}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+266}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-238}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, \mathsf{fma}\left(z, -a, x \cdot y\right)\right)}{t\_1}\\ \mathbf{elif}\;t\_3 \leq 10^{-203}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{+192}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (fma x (/ y (fma z (- b y) y)) (/ (- t a) (- b y))))
        (t_3 (/ (+ (* z (- t a)) (* x y)) t_1)))
   (if (<= t_3 -2e+266)
     t_2
     (if (<= t_3 -1e-238)
       (/ (fma z t (fma z (- a) (* x y))) t_1)
       (if (<= t_3 1e-203) t_2 (if (<= t_3 1e+192) t_3 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 = fma(x, (y / fma(z, (b - y), y)), ((t - a) / (b - y)));
	double t_3 = ((z * (t - a)) + (x * y)) / t_1;
	double tmp;
	if (t_3 <= -2e+266) {
		tmp = t_2;
	} else if (t_3 <= -1e-238) {
		tmp = fma(z, t, fma(z, -a, (x * y))) / t_1;
	} else if (t_3 <= 1e-203) {
		tmp = t_2;
	} else if (t_3 <= 1e+192) {
		tmp = t_3;
	} 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 = fma(x, Float64(y / fma(z, Float64(b - y), y)), Float64(Float64(t - a) / Float64(b - y)))
	t_3 = Float64(Float64(Float64(z * Float64(t - a)) + Float64(x * y)) / t_1)
	tmp = 0.0
	if (t_3 <= -2e+266)
		tmp = t_2;
	elseif (t_3 <= -1e-238)
		tmp = Float64(fma(z, t, fma(z, Float64(-a), Float64(x * y))) / t_1);
	elseif (t_3 <= 1e-203)
		tmp = t_2;
	elseif (t_3 <= 1e+192)
		tmp = t_3;
	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[(x * N[(y / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+266], t$95$2, If[LessEqual[t$95$3, -1e-238], N[(N[(z * t + N[(z * (-a) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 1e-203], t$95$2, If[LessEqual[t$95$3, 1e+192], t$95$3, 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)))) < -2.0000000000000001e266 or -9.9999999999999999e-239 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1e-203 or 1.00000000000000004e192 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 27.6%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 42.8

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

   5. Applied rewrites 42.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 88.7%

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

   if -2.0000000000000001e266 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999999e-239

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 1e-203 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.00000000000000004e192

   1. Initial program 98.4%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq -2 \cdot 10^{+266}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, \frac{t - a}{b - y}\right)\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-238}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, \mathsf{fma}\left(z, -a, x \cdot y\right)\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq 10^{-203}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, \frac{t - a}{b - y}\right)\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq 10^{+192}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}, \frac{t - a}{b - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \mathsf{fma}\left(x, \frac{1}{1 - z}, \frac{t - a}{b - y}\right)\\ t_3 := \frac{z \cdot \left(t - a\right) + x \cdot y}{t\_1}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+307}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-238}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, \mathsf{fma}\left(z, -a, x \cdot y\right)\right)}{t\_1}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{t \cdot \left(1 - \frac{a}{t}\right)}{b - y}\\ \mathbf{elif}\;t\_3 \leq 10^{+192}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (fma x (/ 1.0 (- 1.0 z)) (/ (- t a) (- b y))))
        (t_3 (/ (+ (* z (- t a)) (* x y)) t_1)))
   (if (<= t_3 -1e+307)
     t_2
     (if (<= t_3 -1e-238)
       (/ (fma z t (fma z (- a) (* x y))) t_1)
       (if (<= t_3 0.0)
         (/ (* t (- 1.0 (/ a t))) (- b y))
         (if (<= t_3 1e+192) t_3 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 = fma(x, (1.0 / (1.0 - z)), ((t - a) / (b - y)));
	double t_3 = ((z * (t - a)) + (x * y)) / t_1;
	double tmp;
	if (t_3 <= -1e+307) {
		tmp = t_2;
	} else if (t_3 <= -1e-238) {
		tmp = fma(z, t, fma(z, -a, (x * y))) / t_1;
	} else if (t_3 <= 0.0) {
		tmp = (t * (1.0 - (a / t))) / (b - y);
	} else if (t_3 <= 1e+192) {
		tmp = t_3;
	} 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 = fma(x, Float64(1.0 / Float64(1.0 - z)), Float64(Float64(t - a) / Float64(b - y)))
	t_3 = Float64(Float64(Float64(z * Float64(t - a)) + Float64(x * y)) / t_1)
	tmp = 0.0
	if (t_3 <= -1e+307)
		tmp = t_2;
	elseif (t_3 <= -1e-238)
		tmp = Float64(fma(z, t, fma(z, Float64(-a), Float64(x * y))) / t_1);
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(t * Float64(1.0 - Float64(a / t))) / Float64(b - y));
	elseif (t_3 <= 1e+192)
		tmp = t_3;
	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[(x * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+307], t$95$2, If[LessEqual[t$95$3, -1e-238], N[(N[(z * t + N[(z * (-a) + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(t * N[(1.0 - N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+192], t$95$3, t$95$2]]]]]]]

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)))) < -9.99999999999999986e306 or 1.00000000000000004e192 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 23.1%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 43.1

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

   5. Applied rewrites 43.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 95.1%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 93.3%

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

   if -9.99999999999999986e306 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999999e-239

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if -9.9999999999999999e-239 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 34.0

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

   4. Applied rewrites 34.0%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b + -1 \cdot y}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 64.3

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

   7. Applied rewrites 64.3%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 64.3%

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

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.00000000000000004e192

   1. Initial program 98.4%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{1 - z}, \frac{t - a}{b - y}\right)\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-238}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, \mathsf{fma}\left(z, -a, x \cdot y\right)\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{t \cdot \left(1 - \frac{a}{t}\right)}{b - y}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq 10^{+192}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{1 - z}, \frac{t - a}{b - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \frac{1}{1 - z}, \frac{t - a}{b - y}\right)\\ t_2 := z \cdot \left(t - a\right) + x \cdot y\\ t_3 := \frac{t\_2}{y + z \cdot \left(b - y\right)}\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{+307}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-238}:\\ \;\;\;\;\frac{t\_2}{\mathsf{fma}\left(z, b, \mathsf{fma}\left(z, -y, y\right)\right)}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{t \cdot \left(1 - \frac{a}{t}\right)}{b - y}\\ \mathbf{elif}\;t\_3 \leq 10^{+192}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma x (/ 1.0 (- 1.0 z)) (/ (- t a) (- b y))))
        (t_2 (+ (* z (- t a)) (* x y)))
        (t_3 (/ t_2 (+ y (* z (- b y))))))
   (if (<= t_3 -1e+307)
     t_1
     (if (<= t_3 -1e-238)
       (/ t_2 (fma z b (fma z (- y) y)))
       (if (<= t_3 0.0)
         (/ (* t (- 1.0 (/ a t))) (- b y))
         (if (<= t_3 1e+192) t_3 t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(x, (1.0 / (1.0 - z)), ((t - a) / (b - y)));
	double t_2 = (z * (t - a)) + (x * y);
	double t_3 = t_2 / (y + (z * (b - y)));
	double tmp;
	if (t_3 <= -1e+307) {
		tmp = t_1;
	} else if (t_3 <= -1e-238) {
		tmp = t_2 / fma(z, b, fma(z, -y, y));
	} else if (t_3 <= 0.0) {
		tmp = (t * (1.0 - (a / t))) / (b - y);
	} else if (t_3 <= 1e+192) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(x, Float64(1.0 / Float64(1.0 - z)), Float64(Float64(t - a) / Float64(b - y)))
	t_2 = Float64(Float64(z * Float64(t - a)) + Float64(x * y))
	t_3 = Float64(t_2 / Float64(y + Float64(z * Float64(b - y))))
	tmp = 0.0
	if (t_3 <= -1e+307)
		tmp = t_1;
	elseif (t_3 <= -1e-238)
		tmp = Float64(t_2 / fma(z, b, fma(z, Float64(-y), y)));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(t * Float64(1.0 - Float64(a / t))) / Float64(b - y));
	elseif (t_3 <= 1e+192)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+307], t$95$1, If[LessEqual[t$95$3, -1e-238], N[(t$95$2 / N[(z * b + N[(z * (-y) + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(t * N[(1.0 - N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 1e+192], t$95$3, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.99999999999999986e306 or 1.00000000000000004e192 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 23.1%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 43.1

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

   5. Applied rewrites 43.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 95.1%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 93.3%

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

   if -9.99999999999999986e306 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999999e-239

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if -9.9999999999999999e-239 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 34.0

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

   4. Applied rewrites 34.0%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b + -1 \cdot y}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 64.3

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

   7. Applied rewrites 64.3%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 64.3%

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

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.00000000000000004e192

   1. Initial program 98.4%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{1 - z}, \frac{t - a}{b - y}\right)\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-238}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + x \cdot y}{\mathsf{fma}\left(z, b, \mathsf{fma}\left(z, -y, y\right)\right)}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{t \cdot \left(1 - \frac{a}{t}\right)}{b - y}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq 10^{+192}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{1 - z}, \frac{t - a}{b - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \frac{1}{1 - z}, \frac{t - a}{b - y}\right)\\ t_2 := \frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+307}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-238}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{t \cdot \left(1 - \frac{a}{t}\right)}{b - y}\\ \mathbf{elif}\;t\_2 \leq 10^{+192}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma x (/ 1.0 (- 1.0 z)) (/ (- t a) (- b y))))
        (t_2 (/ (+ (* z (- t a)) (* x y)) (+ y (* z (- b y))))))
   (if (<= t_2 -1e+307)
     t_1
     (if (<= t_2 -1e-238)
       t_2
       (if (<= t_2 0.0)
         (/ (* t (- 1.0 (/ a t))) (- b y))
         (if (<= t_2 1e+192) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(x, (1.0 / (1.0 - z)), ((t - a) / (b - y)));
	double t_2 = ((z * (t - a)) + (x * y)) / (y + (z * (b - y)));
	double tmp;
	if (t_2 <= -1e+307) {
		tmp = t_1;
	} else if (t_2 <= -1e-238) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = (t * (1.0 - (a / t))) / (b - y);
	} else if (t_2 <= 1e+192) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(x, Float64(1.0 / Float64(1.0 - z)), Float64(Float64(t - a) / Float64(b - y)))
	t_2 = Float64(Float64(Float64(z * Float64(t - a)) + Float64(x * y)) / Float64(y + Float64(z * Float64(b - y))))
	tmp = 0.0
	if (t_2 <= -1e+307)
		tmp = t_1;
	elseif (t_2 <= -1e-238)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(t * Float64(1.0 - Float64(a / t))) / Float64(b - y));
	elseif (t_2 <= 1e+192)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+307], t$95$1, If[LessEqual[t$95$2, -1e-238], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(t * N[(1.0 - N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+192], t$95$2, 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)))) < -9.99999999999999986e306 or 1.00000000000000004e192 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 23.1%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 43.1

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

   5. Applied rewrites 43.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 95.1%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 93.3%

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

   if -9.99999999999999986e306 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999999e-239 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.00000000000000004e192

   1. Initial program 99.1%

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

   if -9.9999999999999999e-239 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 34.0

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

   4. Applied rewrites 34.0%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b + -1 \cdot y}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 64.3

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

   7. Applied rewrites 64.3%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 64.3%

       \[\leadsto \frac{t \cdot \left(1 - \frac{a}{t}\right)}{\color{blue}{b} - y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{1 - z}, \frac{t - a}{b - y}\right)\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-238}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{t \cdot \left(1 - \frac{a}{t}\right)}{b - y}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq 10^{+192}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{1 - z}, \frac{t - a}{b - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)}\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \mathsf{fma}\left(x, \frac{1}{1 - z}, t\_2\right)\\ t_4 := \frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+173}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-238}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 10^{-203}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+70}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* z (- t a)) (* x y)) (+ y (* z (- b y)))))
        (t_2 (/ (- t a) (- b y)))
        (t_3 (fma x (/ 1.0 (- 1.0 z)) t_2))
        (t_4 (/ (fma z t (* x y)) (fma z (- b y) y))))
   (if (<= t_1 -1e+173)
     t_3
     (if (<= t_1 -1e-238)
       t_4
       (if (<= t_1 1e-203) t_2 (if (<= t_1 5e+70) t_4 t_3))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((z * (t - a)) + (x * y)) / (y + (z * (b - y)));
	double t_2 = (t - a) / (b - y);
	double t_3 = fma(x, (1.0 / (1.0 - z)), t_2);
	double t_4 = fma(z, t, (x * y)) / fma(z, (b - y), y);
	double tmp;
	if (t_1 <= -1e+173) {
		tmp = t_3;
	} else if (t_1 <= -1e-238) {
		tmp = t_4;
	} else if (t_1 <= 1e-203) {
		tmp = t_2;
	} else if (t_1 <= 5e+70) {
		tmp = t_4;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(z * Float64(t - a)) + Float64(x * y)) / Float64(y + Float64(z * Float64(b - y))))
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	t_3 = fma(x, Float64(1.0 / Float64(1.0 - z)), t_2)
	t_4 = Float64(fma(z, t, Float64(x * y)) / fma(z, Float64(b - y), y))
	tmp = 0.0
	if (t_1 <= -1e+173)
		tmp = t_3;
	elseif (t_1 <= -1e-238)
		tmp = t_4;
	elseif (t_1 <= 1e-203)
		tmp = t_2;
	elseif (t_1 <= 5e+70)
		tmp = t_4;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+173], t$95$3, If[LessEqual[t$95$1, -1e-238], t$95$4, If[LessEqual[t$95$1, 1e-203], t$95$2, If[LessEqual[t$95$1, 5e+70], t$95$4, t$95$3]]]]]]]]

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)))) < -1e173 or 5.0000000000000002e70 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 36.0%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 52.7

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 92.1%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 89.9%

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

   if -1e173 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -9.9999999999999999e-239 or 1e-203 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 5.0000000000000002e70

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 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 79.4

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

   5. Applied rewrites 79.4%

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

   if -9.9999999999999999e-239 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1e-203

   1. Initial program 40.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 64.7

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

   5. Applied rewrites 64.7%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{+173}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{1 - z}, \frac{t - a}{b - y}\right)\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-238}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq 10^{-203}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)} \leq 5 \cdot 10^{+70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{1 - z}, \frac{t - a}{b - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \frac{-1}{z}, \frac{t - a}{b - y}\right)\\ \mathbf{if}\;z \leq -5.1 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma x (/ -1.0 z) (/ (- t a) (- b y)))))
   (if (<= z -5.1e-6)
     t_1
     (if (<= z 9.5e-9) (/ (fma z t (* x y)) (fma z (- b y) y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(x, (-1.0 / z), ((t - a) / (b - y)));
	double tmp;
	if (z <= -5.1e-6) {
		tmp = t_1;
	} else if (z <= 9.5e-9) {
		tmp = fma(z, t, (x * y)) / fma(z, (b - y), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(x, Float64(-1.0 / z), Float64(Float64(t - a) / Float64(b - y)))
	tmp = 0.0
	if (z <= -5.1e-6)
		tmp = t_1;
	elseif (z <= 9.5e-9)
		tmp = Float64(fma(z, t, Float64(x * y)) / fma(z, Float64(b - y), y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x * N[(-1.0 / z), $MachinePrecision] + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.1e-6], t$95$1, If[LessEqual[z, 9.5e-9], N[(N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.1000000000000003e-6 or 9.5000000000000007e-9 < z

   1. Initial program 38.7%

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

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 43.0

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

   5. Applied rewrites 43.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 88.5%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 87.0%

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

   12. Taylor expanded in z around inf

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

   14. Applied rewrites 86.3%

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

   if -5.1000000000000003e-6 < z < 9.5000000000000007e-9

   1. Initial program 86.2%

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

   3. Taylor expanded in 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 67.6

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

   5. Applied rewrites 67.6%

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

Alternative 8: 68.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, b - y, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -960:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-175}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right)}{t\_1}\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{y}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -960.0)
     t_2
     (if (<= z -3.1e-175)
       (/ (* z (- t a)) t_1)
       (if (<= z 9.8e-14) (* x (/ y 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 <= -960.0) {
		tmp = t_2;
	} else if (z <= -3.1e-175) {
		tmp = (z * (t - a)) / t_1;
	} else if (z <= 9.8e-14) {
		tmp = x * (y / 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 <= -960.0)
		tmp = t_2;
	elseif (z <= -3.1e-175)
		tmp = Float64(Float64(z * Float64(t - a)) / t_1);
	elseif (z <= 9.8e-14)
		tmp = Float64(x * Float64(y / 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, -960.0], t$95$2, If[LessEqual[z, -3.1e-175], N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 9.8e-14], N[(x * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -960 or 9.79999999999999989e-14 < z

   1. Initial program 37.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if -960 < z < -3.09999999999999999e-175

   1. Initial program 92.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 66.8

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

   5. Applied rewrites 66.8%

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

   if -3.09999999999999999e-175 < z < 9.79999999999999989e-14

   1. Initial program 83.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 83.2

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

   4. Applied rewrites 83.2%

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

   5. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 72.8

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

   7. Applied rewrites 72.8%

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

Alternative 9: 71.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1e-56)
     t_1
     (if (<= z 1.4e-13) (/ (fma z t (* x y)) (fma z (- b y) y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1e-56) {
		tmp = t_1;
	} else if (z <= 1.4e-13) {
		tmp = fma(z, t, (x * y)) / fma(z, (b - y), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1e-56)
		tmp = t_1;
	elseif (z <= 1.4e-13)
		tmp = Float64(fma(z, t, Float64(x * y)) / fma(z, Float64(b - y), y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e-56], t$95$1, If[LessEqual[z, 1.4e-13], N[(N[(z * t + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1e-56 or 1.4000000000000001e-13 < z

   1. Initial program 41.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 78.4

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

   5. Applied rewrites 78.4%

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

   if -1e-56 < z < 1.4000000000000001e-13

   1. Initial program 86.3%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 67.3

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

   5. Applied rewrites 67.3%

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

Alternative 10: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{-69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -6.5e-69)
     t_1
     (if (<= z 9.8e-14) (* x (/ y (fma z (- b y) y))) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -6.5e-69) {
		tmp = t_1;
	} else if (z <= 9.8e-14) {
		tmp = x * (y / fma(z, (b - y), y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -6.5e-69)
		tmp = t_1;
	elseif (z <= 9.8e-14)
		tmp = Float64(x * Float64(y / fma(z, Float64(b - y), y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.5e-69], t$95$1, If[LessEqual[z, 9.8e-14], N[(x * N[(y / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.49999999999999951e-69 or 9.79999999999999989e-14 < z

   1. Initial program 43.0%

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

   3. Taylor expanded in 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 76.6

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

   5. Applied rewrites 76.6%

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

   if -6.49999999999999951e-69 < z < 9.79999999999999989e-14

   1. Initial program 85.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 85.6

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

   4. Applied rewrites 85.6%

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

   5. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 65.2

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

   7. Applied rewrites 65.2%

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

Alternative 11: 63.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -6 \cdot 10^{-69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-297}:\\ \;\;\;\;\frac{x \cdot y}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -6e-69)
     t_1
     (if (<= z -6.5e-297)
       (/ (* x y) (fma z (- b y) y))
       (if (<= z 3.4e-16) (fma z x 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 <= -6e-69) {
		tmp = t_1;
	} else if (z <= -6.5e-297) {
		tmp = (x * y) / fma(z, (b - y), y);
	} else if (z <= 3.4e-16) {
		tmp = fma(z, x, 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 <= -6e-69)
		tmp = t_1;
	elseif (z <= -6.5e-297)
		tmp = Float64(Float64(x * y) / fma(z, Float64(b - y), y));
	elseif (z <= 3.4e-16)
		tmp = fma(z, x, 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, -6e-69], t$95$1, If[LessEqual[z, -6.5e-297], N[(N[(x * y), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.4e-16], N[(z * x + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.99999999999999978e-69 or 3.4e-16 < z

   1. Initial program 43.0%

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

   3. Taylor expanded in 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 76.6

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

   5. Applied rewrites 76.6%

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

   if -5.99999999999999978e-69 < z < -6.5000000000000002e-297

   1. Initial program 94.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 58.8

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

   5. Applied rewrites 58.8%

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

   if -6.5000000000000002e-297 < z < 3.4e-16

   1. Initial program 77.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 64.9

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

   5. Applied rewrites 64.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 64.9%

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

Alternative 12: 53.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -2.55 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{elif}\;y \leq 300000:\\ \;\;\;\;\frac{t - a}{-y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -2.55e+23)
     t_1
     (if (<= y 6.5e-87)
       (/ (- t a) b)
       (if (<= y 300000.0) (/ (- t a) (- y)) 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 <= -2.55e+23) {
		tmp = t_1;
	} else if (y <= 6.5e-87) {
		tmp = (t - a) / b;
	} else if (y <= 300000.0) {
		tmp = (t - a) / -y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (1.0d0 - z)
    if (y <= (-2.55d+23)) then
        tmp = t_1
    else if (y <= 6.5d-87) then
        tmp = (t - a) / b
    else if (y <= 300000.0d0) then
        tmp = (t - a) / -y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -2.55e+23) {
		tmp = t_1;
	} else if (y <= 6.5e-87) {
		tmp = (t - a) / b;
	} else if (y <= 300000.0) {
		tmp = (t - a) / -y;
	} 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 <= -2.55e+23:
		tmp = t_1
	elif y <= 6.5e-87:
		tmp = (t - a) / b
	elif y <= 300000.0:
		tmp = (t - a) / -y
	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 <= -2.55e+23)
		tmp = t_1;
	elseif (y <= 6.5e-87)
		tmp = Float64(Float64(t - a) / b);
	elseif (y <= 300000.0)
		tmp = Float64(Float64(t - a) / Float64(-y));
	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 <= -2.55e+23)
		tmp = t_1;
	elseif (y <= 6.5e-87)
		tmp = (t - a) / b;
	elseif (y <= 300000.0)
		tmp = (t - a) / -y;
	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, -2.55e+23], t$95$1, If[LessEqual[y, 6.5e-87], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[y, 300000.0], N[(N[(t - a), $MachinePrecision] / (-y)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5500000000000001e23 or 3e5 < y

   1. Initial program 47.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 49.5

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

   5. Applied rewrites 49.5%

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

   if -2.5500000000000001e23 < y < 6.5000000000000003e-87

   1. Initial program 77.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 64.2

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

   5. Applied rewrites 64.2%

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

   if 6.5000000000000003e-87 < y < 3e5

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 76.6

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b + -1 \cdot y}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 53.1

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

   7. Applied rewrites 53.1%

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

   8. Taylor expanded in b around 0

      \[\leadsto \frac{t - a}{-1 \cdot \color{blue}{y}} \]
   9. Step-by-step derivation

   10. Applied rewrites 43.0%

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

Alternative 13: 62.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{-177}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.6e-177) t_1 (if (<= z 3.4e-16) (fma z x x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.6e-177) {
		tmp = t_1;
	} else if (z <= 3.4e-16) {
		tmp = fma(z, x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.6e-177)
		tmp = t_1;
	elseif (z <= 3.4e-16)
		tmp = fma(z, x, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.6e-177], t$95$1, If[LessEqual[z, 3.4e-16], N[(z * x + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5999999999999999e-177 or 3.4e-16 < z

   1. Initial program 51.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 69.1

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

   5. Applied rewrites 69.1%

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

   if -1.5999999999999999e-177 < z < 3.4e-16

   1. Initial program 83.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 65.6

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

   5. Applied rewrites 65.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 65.6%

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

Alternative 14: 54.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -2.55 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+39}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -2.55e+23) t_1 (if (<= y 4.1e+39) (/ (- 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 <= -2.55e+23) {
		tmp = t_1;
	} else if (y <= 4.1e+39) {
		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 <= (-2.55d+23)) then
        tmp = t_1
    else if (y <= 4.1d+39) 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 <= -2.55e+23) {
		tmp = t_1;
	} else if (y <= 4.1e+39) {
		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 <= -2.55e+23:
		tmp = t_1
	elif y <= 4.1e+39:
		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 <= -2.55e+23)
		tmp = t_1;
	elseif (y <= 4.1e+39)
		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 <= -2.55e+23)
		tmp = t_1;
	elseif (y <= 4.1e+39)
		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, -2.55e+23], t$95$1, If[LessEqual[y, 4.1e+39], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5500000000000001e23 or 4.10000000000000004e39 < y

   1. Initial program 45.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 50.0

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

   5. Applied rewrites 50.0%

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

   if -2.5500000000000001e23 < y < 4.10000000000000004e39

   1. Initial program 76.5%

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

   3. Taylor expanded in y around 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 55.9

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

   5. Applied rewrites 55.9%

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

Alternative 15: 42.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{y - b}\\ \mathbf{if}\;a \leq -3.75 \cdot 10^{+115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+116}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (- y b))))
   (if (<= a -3.75e+115) t_1 (if (<= a 2.2e+116) (/ x (- 1.0 z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (y - b);
	double tmp;
	if (a <= -3.75e+115) {
		tmp = t_1;
	} else if (a <= 2.2e+116) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / (y - b)
    if (a <= (-3.75d+115)) then
        tmp = t_1
    else if (a <= 2.2d+116) then
        tmp = x / (1.0d0 - z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (y - b);
	double tmp;
	if (a <= -3.75e+115) {
		tmp = t_1;
	} else if (a <= 2.2e+116) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (y - b)
	tmp = 0
	if a <= -3.75e+115:
		tmp = t_1
	elif a <= 2.2e+116:
		tmp = x / (1.0 - z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(y - b))
	tmp = 0.0
	if (a <= -3.75e+115)
		tmp = t_1;
	elseif (a <= 2.2e+116)
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (y - b);
	tmp = 0.0;
	if (a <= -3.75e+115)
		tmp = t_1;
	elseif (a <= 2.2e+116)
		tmp = x / (1.0 - z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.75e+115], t$95$1, If[LessEqual[a, 2.2e+116], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.7499999999999998e115 or 2.2e116 < a

   1. Initial program 56.4%

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

   3. Taylor expanded in a around 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 39.0

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

   5. Applied rewrites 39.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 59.8%

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

   if -3.7499999999999998e115 < a < 2.2e116

   1. Initial program 63.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 42.8

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

   5. Applied rewrites 42.8%

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

Alternative 16: 45.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{y - b}\\ \mathbf{if}\;z \leq -1.95 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-14}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (- y b))))
   (if (<= z -1.95e-8) t_1 (if (<= z 9.8e-14) (* x 1.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (y - b);
	double tmp;
	if (z <= -1.95e-8) {
		tmp = t_1;
	} else if (z <= 9.8e-14) {
		tmp = x * 1.0;
	} 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 = a / (y - b)
    if (z <= (-1.95d-8)) then
        tmp = t_1
    else if (z <= 9.8d-14) then
        tmp = x * 1.0d0
    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 = a / (y - b);
	double tmp;
	if (z <= -1.95e-8) {
		tmp = t_1;
	} else if (z <= 9.8e-14) {
		tmp = x * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (y - b)
	tmp = 0
	if z <= -1.95e-8:
		tmp = t_1
	elif z <= 9.8e-14:
		tmp = x * 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(y - b))
	tmp = 0.0
	if (z <= -1.95e-8)
		tmp = t_1;
	elseif (z <= 9.8e-14)
		tmp = Float64(x * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (y - b);
	tmp = 0.0;
	if (z <= -1.95e-8)
		tmp = t_1;
	elseif (z <= 9.8e-14)
		tmp = x * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.95e-8], t$95$1, If[LessEqual[z, 9.8e-14], N[(x * 1.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.94999999999999992e-8 or 9.79999999999999989e-14 < z

   1. Initial program 39.1%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. 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 20.5

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

   5. Applied rewrites 20.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 43.8%

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

   if -1.94999999999999992e-8 < z < 9.79999999999999989e-14

   1. Initial program 86.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 86.1

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

   4. Applied rewrites 86.1%

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

   5. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 61.2

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

   7. Applied rewrites 61.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 52.6%

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

Alternative 17: 36.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{-b}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-14}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (- b))))
   (if (<= z -1e-56) t_1 (if (<= z 9.8e-14) (* x 1.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / -b;
	double tmp;
	if (z <= -1e-56) {
		tmp = t_1;
	} else if (z <= 9.8e-14) {
		tmp = x * 1.0;
	} 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 = a / -b
    if (z <= (-1d-56)) then
        tmp = t_1
    else if (z <= 9.8d-14) then
        tmp = x * 1.0d0
    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 = a / -b;
	double tmp;
	if (z <= -1e-56) {
		tmp = t_1;
	} else if (z <= 9.8e-14) {
		tmp = x * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / -b
	tmp = 0
	if z <= -1e-56:
		tmp = t_1
	elif z <= 9.8e-14:
		tmp = x * 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(-b))
	tmp = 0.0
	if (z <= -1e-56)
		tmp = t_1;
	elseif (z <= 9.8e-14)
		tmp = Float64(x * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / -b;
	tmp = 0.0;
	if (z <= -1e-56)
		tmp = t_1;
	elseif (z <= 9.8e-14)
		tmp = x * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / (-b)), $MachinePrecision]}, If[LessEqual[z, -1e-56], t$95$1, If[LessEqual[z, 9.8e-14], N[(x * 1.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1e-56 or 9.79999999999999989e-14 < z

   1. Initial program 41.0%

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

   3. Taylor expanded in a around inf

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

      1. 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 20.5

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

   5. Applied rewrites 20.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 22.0%

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

   if -1e-56 < z < 9.79999999999999989e-14

   1. Initial program 86.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 86.3

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

   4. Applied rewrites 86.3%

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

   5. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower--.f64 63.4

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

   7. Applied rewrites 63.4%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 54.3%

       \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-56}:\\ \;\;\;\;\frac{a}{-b}\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-14}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 25.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. 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 34.9

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

5. Applied rewrites 34.9%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 27.0%

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

Alternative 19: 25.2% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. associate-+l+ N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 61.1

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

4. Applied rewrites 61.1%

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

5. Taylor expanded in x around inf

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

   1. associate-/l* N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-in N/A

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

   5. mul-1-neg N/A

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

   6. sub-neg N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower--.f64 37.5

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

7. Applied rewrites 37.5%

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

8. Taylor expanded in z around 0

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

10. Applied rewrites 26.7%

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

Alternative 20: 3.9% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

3. Taylor expanded in y around inf

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

   1. lower-/.f64 N/A

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

   2. 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 34.9

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

5. Applied rewrites 34.9%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 26.5%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 4.0%

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

Developer Target 1: 73.6% 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 2024233 
(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)))))