Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.0% → 95.7%

Time: 13.4s

Alternatives: 18

Speedup: 1.1×

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: 66.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{z}, \frac{y}{b - y}, \frac{t}{b - y}\right) - \mathsf{fma}\left(\frac{y}{z}, \frac{t - a}{{\left(b - y\right)}^{2}}, \frac{a}{b - y}\right)\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ t_3 := \frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ t_4 := \mathsf{fma}\left(\frac{z}{x}, \frac{t - a}{t\_2}, \frac{y}{t\_2}\right) \cdot x\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-260}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+254}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (-
          (fma (/ x z) (/ y (- b y)) (/ t (- b y)))
          (fma (/ y z) (/ (- t a) (pow (- b y) 2.0)) (/ a (- b y)))))
        (t_2 (fma (- b y) z y))
        (t_3 (/ (+ (* (- t a) z) (* y x)) (+ (* (- b y) z) y)))
        (t_4 (* (fma (/ z x) (/ (- t a) t_2) (/ y t_2)) x)))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -5e-260)
       t_3
       (if (<= t_3 0.0)
         t_1
         (if (<= t_3 5e+254) t_3 (if (<= t_3 INFINITY) t_4 t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((x / z), (y / (b - y)), (t / (b - y))) - fma((y / z), ((t - a) / pow((b - y), 2.0)), (a / (b - y)));
	double t_2 = fma((b - y), z, y);
	double t_3 = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	double t_4 = fma((z / x), ((t - a) / t_2), (y / t_2)) * x;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -5e-260) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = t_1;
	} else if (t_3 <= 5e+254) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(x / z), Float64(y / Float64(b - y)), Float64(t / Float64(b - y))) - fma(Float64(y / z), Float64(Float64(t - a) / (Float64(b - y) ^ 2.0)), Float64(a / Float64(b - y))))
	t_2 = fma(Float64(b - y), z, y)
	t_3 = Float64(Float64(Float64(Float64(t - a) * z) + Float64(y * x)) / Float64(Float64(Float64(b - y) * z) + y))
	t_4 = Float64(fma(Float64(z / x), Float64(Float64(t - a) / t_2), Float64(y / t_2)) * x)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -5e-260)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = t_1;
	elseif (t_3 <= 5e+254)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(x / z), $MachinePrecision] * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y / z), $MachinePrecision] * N[(N[(t - a), $MachinePrecision] / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(z / x), $MachinePrecision] * N[(N[(t - a), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(y / t$95$2), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -5e-260], t$95$3, If[LessEqual[t$95$3, 0.0], t$95$1, If[LessEqual[t$95$3, 5e+254], t$95$3, If[LessEqual[t$95$3, Infinity], t$95$4, t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 4.99999999999999994e254 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 28.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 44.7

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 91.5%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -5.0000000000000003e-260 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 4.99999999999999994e254

   1. Initial program 99.7%

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

   if -5.0000000000000003e-260 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 8.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 23.2

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

   5. Applied rewrites 23.2%

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

   6. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

   8. Applied rewrites 100.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{x}, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, \frac{y}{\mathsf{fma}\left(b - y, z, y\right)}\right) \cdot x\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq -5 \cdot 10^{-260}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \frac{y}{b - y}, \frac{t}{b - y}\right) - \mathsf{fma}\left(\frac{y}{z}, \frac{t - a}{{\left(b - y\right)}^{2}}, \frac{a}{b - y}\right)\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq 5 \cdot 10^{+254}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{x}, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, \frac{y}{\mathsf{fma}\left(b - y, z, y\right)}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, \frac{y}{b - y}, \frac{t}{b - y}\right) - \mathsf{fma}\left(\frac{y}{z}, \frac{t - a}{{\left(b - y\right)}^{2}}, \frac{a}{b - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y} - \frac{\mathsf{fma}\left(-y, \frac{x}{b - y}, \frac{y}{{\left(b - y\right)}^{2}} \cdot \left(t - a\right)\right)}{z}\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ t_3 := \frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ t_4 := \mathsf{fma}\left(\frac{z}{x}, \frac{t - a}{t\_2}, \frac{y}{t\_2}\right) \cdot x\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-260}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+254}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (-
          (/ (- t a) (- b y))
          (/ (fma (- y) (/ x (- b y)) (* (/ y (pow (- b y) 2.0)) (- t a))) z)))
        (t_2 (fma (- b y) z y))
        (t_3 (/ (+ (* (- t a) z) (* y x)) (+ (* (- b y) z) y)))
        (t_4 (* (fma (/ z x) (/ (- t a) t_2) (/ y t_2)) x)))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -5e-260)
       t_3
       (if (<= t_3 0.0)
         t_1
         (if (<= t_3 5e+254) t_3 (if (<= t_3 INFINITY) t_4 t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((t - a) / (b - y)) - (fma(-y, (x / (b - y)), ((y / pow((b - y), 2.0)) * (t - a))) / z);
	double t_2 = fma((b - y), z, y);
	double t_3 = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	double t_4 = fma((z / x), ((t - a) / t_2), (y / t_2)) * x;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -5e-260) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = t_1;
	} else if (t_3 <= 5e+254) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(t - a) / Float64(b - y)) - Float64(fma(Float64(-y), Float64(x / Float64(b - y)), Float64(Float64(y / (Float64(b - y) ^ 2.0)) * Float64(t - a))) / z))
	t_2 = fma(Float64(b - y), z, y)
	t_3 = Float64(Float64(Float64(Float64(t - a) * z) + Float64(y * x)) / Float64(Float64(Float64(b - y) * z) + y))
	t_4 = Float64(fma(Float64(z / x), Float64(Float64(t - a) / t_2), Float64(y / t_2)) * x)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -5e-260)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = t_1;
	elseif (t_3 <= 5e+254)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(N[((-y) * N[(x / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(N[(y / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(z / x), $MachinePrecision] * N[(N[(t - a), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(y / t$95$2), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -5e-260], t$95$3, If[LessEqual[t$95$3, 0.0], t$95$1, If[LessEqual[t$95$3, 5e+254], t$95$3, If[LessEqual[t$95$3, Infinity], t$95$4, t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 4.99999999999999994e254 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 28.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 44.7

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 91.5%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -5.0000000000000003e-260 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 4.99999999999999994e254

   1. Initial program 99.7%

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

   if -5.0000000000000003e-260 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 8.7%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 99.8%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{x}, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, \frac{y}{\mathsf{fma}\left(b - y, z, y\right)}\right) \cdot x\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq -5 \cdot 10^{-260}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq 0:\\ \;\;\;\;\frac{t - a}{b - y} - \frac{\mathsf{fma}\left(-y, \frac{x}{b - y}, \frac{y}{{\left(b - y\right)}^{2}} \cdot \left(t - a\right)\right)}{z}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq 5 \cdot 10^{+254}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{x}, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, \frac{y}{\mathsf{fma}\left(b - y, z, y\right)}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y} - \frac{\mathsf{fma}\left(-y, \frac{x}{b - y}, \frac{y}{{\left(b - y\right)}^{2}} \cdot \left(t - a\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ t_3 := \mathsf{fma}\left(\frac{z}{x}, \frac{t - a}{t\_1}, \frac{y}{t\_1}\right) \cdot x\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+254}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y))
        (t_2 (/ (+ (* (- t a) z) (* y x)) (+ (* (- b y) z) y)))
        (t_3 (* (fma (/ z x) (/ (- t a) t_1) (/ y t_1)) x)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 5e+254) t_2 (if (<= t_2 INFINITY) t_3 (/ (- t a) (- b y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	double t_3 = fma((z / x), ((t - a) / t_1), (y / t_1)) * x;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 5e+254) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(Float64(Float64(t - a) * z) + Float64(y * x)) / Float64(Float64(Float64(b - y) * z) + y))
	t_3 = Float64(fma(Float64(z / x), Float64(Float64(t - a) / t_1), Float64(y / t_1)) * x)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 5e+254)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(z / x), $MachinePrecision] * N[(N[(t - a), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(y / t$95$1), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, 5e+254], t$95$2, If[LessEqual[t$95$2, Infinity], t$95$3, N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 4.99999999999999994e254 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 28.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 44.7

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 91.5%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 4.99999999999999994e254

   1. Initial program 93.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 72.5

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

   5. Applied rewrites 72.5%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{x}, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, \frac{y}{\mathsf{fma}\left(b - y, z, y\right)}\right) \cdot x\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq 5 \cdot 10^{+254}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ \mathbf{elif}\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{x}, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, \frac{y}{\mathsf{fma}\left(b - y, z, y\right)}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* (- t a) z) (* y x)) (+ (* (- b y) z) y))))
   (if (<= t_1 (- INFINITY))
     (* (fma (/ z x) (/ (- t a) (fma (- b y) z y)) 1.0) x)
     (if (<= t_1 5e+277) t_1 (/ (- t a) (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((z / x), ((t - a) / fma((b - y), z, y)), 1.0) * x;
	} else if (t_1 <= 5e+277) {
		tmp = t_1;
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(t - a) * z) + Float64(y * x)) / Float64(Float64(Float64(b - y) * z) + y))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(fma(Float64(z / x), Float64(Float64(t - a) / fma(Float64(b - y), z, y)), 1.0) * x);
	elseif (t_1 <= 5e+277)
		tmp = t_1;
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 28.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 95.1%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 80.7%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 4.99999999999999982e277

   1. Initial program 93.7%

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

   if 4.99999999999999982e277 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 9.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 69.5

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

   5. Applied rewrites 69.5%

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

4. Final simplification 86.7%

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

Alternative 5: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+61}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -2.2e+25)
     t_1
     (if (<= z 1.1e+61)
       (/ (+ (* (- t a) z) (* y x)) (+ (* (- b y) z) y))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -2.2e+25) {
		tmp = t_1;
	} else if (z <= 1.1e+61) {
		tmp = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-2.2d+25)) then
        tmp = t_1
    else if (z <= 1.1d+61) then
        tmp = (((t - a) * z) + (y * x)) / (((b - y) * z) + y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -2.2e+25) {
		tmp = t_1;
	} else if (z <= 1.1e+61) {
		tmp = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -2.2e+25:
		tmp = t_1
	elif z <= 1.1e+61:
		tmp = (((t - a) * z) + (y * x)) / (((b - y) * z) + y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -2.2e+25)
		tmp = t_1;
	elseif (z <= 1.1e+61)
		tmp = Float64(Float64(Float64(Float64(t - a) * z) + Float64(y * x)) / Float64(Float64(Float64(b - y) * z) + y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -2.2e+25)
		tmp = t_1;
	elseif (z <= 1.1e+61)
		tmp = (((t - a) * z) + (y * x)) / (((b - y) * z) + y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.2e+25], t$95$1, If[LessEqual[z, 1.1e+61], N[(N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2000000000000001e25 or 1.1e61 < z

   1. Initial program 34.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -2.2000000000000001e25 < z < 1.1e61

   1. Initial program 88.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+61}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z + y \cdot x}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4.6 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-125}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t - a, \frac{x}{1 - z}\right)\\ \mathbf{elif}\;z \leq 78000000000:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4.6e-20)
     t_1
     (if (<= z 1.02e-125)
       (fma (/ z y) (- t a) (/ x (- 1.0 z)))
       (if (<= z 78000000000.0) (/ (* (- t a) z) (+ (* (- b y) z) y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -4.6e-20) {
		tmp = t_1;
	} else if (z <= 1.02e-125) {
		tmp = fma((z / y), (t - a), (x / (1.0 - z)));
	} else if (z <= 78000000000.0) {
		tmp = ((t - a) * z) / (((b - y) * z) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -4.6e-20)
		tmp = t_1;
	elseif (z <= 1.02e-125)
		tmp = fma(Float64(z / y), Float64(t - a), Float64(x / Float64(1.0 - z)));
	elseif (z <= 78000000000.0)
		tmp = Float64(Float64(Float64(t - a) * z) / Float64(Float64(Float64(b - y) * z) + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.6e-20], t$95$1, If[LessEqual[z, 1.02e-125], N[(N[(z / y), $MachinePrecision] * N[(t - a), $MachinePrecision] + N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 78000000000.0], N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.5999999999999998e-20 or 7.8e10 < z

   1. Initial program 45.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 78.9

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

   5. Applied rewrites 78.9%

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

   if -4.5999999999999998e-20 < z < 1.02e-125

   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 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 62.8

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.9

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

   8. Applied rewrites 67.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 78.1%

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

   12. Taylor expanded in z around 0

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

   14. Applied rewrites 78.1%

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

   if 1.02e-125 < z < 7.8e10

   1. Initial program 92.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 77.2

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

   5. Applied rewrites 77.2%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-125}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t - a, \frac{x}{1 - z}\right)\\ \mathbf{elif}\;z \leq 78000000000:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4.6 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.02 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{y} + x, z, x\right)\\ \mathbf{elif}\;z \leq 78000000000:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4.6e-20)
     t_1
     (if (<= z 2.02e-100)
       (fma (+ (/ (- t a) y) x) z x)
       (if (<= z 78000000000.0) (/ (* (- t a) z) (+ (* (- b y) z) y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -4.6e-20) {
		tmp = t_1;
	} else if (z <= 2.02e-100) {
		tmp = fma((((t - a) / y) + x), z, x);
	} else if (z <= 78000000000.0) {
		tmp = ((t - a) * z) / (((b - y) * z) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -4.6e-20)
		tmp = t_1;
	elseif (z <= 2.02e-100)
		tmp = fma(Float64(Float64(Float64(t - a) / y) + x), z, x);
	elseif (z <= 78000000000.0)
		tmp = Float64(Float64(Float64(t - a) * z) / Float64(Float64(Float64(b - y) * z) + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.6e-20], t$95$1, If[LessEqual[z, 2.02e-100], N[(N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[z, 78000000000.0], N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.5999999999999998e-20 or 7.8e10 < z

   1. Initial program 45.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 78.9

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

   5. Applied rewrites 78.9%

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

   if -4.5999999999999998e-20 < z < 2.02000000000000005e-100

   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. 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 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 67.7

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

   8. Applied rewrites 67.7%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 77.5%

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

   12. Taylor expanded in z around 0

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

   14. Applied rewrites 73.0%

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

   if 2.02000000000000005e-100 < z < 7.8e10

   1. Initial program 94.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 76.5

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

   5. Applied rewrites 76.5%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 2.02 \cdot 10^{-100}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{y} + x, z, x\right)\\ \mathbf{elif}\;z \leq 78000000000:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 10^{-125}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4.2e-92)
     t_1
     (if (<= z 1e-125)
       (fma z x x)
       (if (<= z 1.1e-62) (/ (* (- t a) z) y) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -4.2e-92) {
		tmp = t_1;
	} else if (z <= 1e-125) {
		tmp = fma(z, x, x);
	} else if (z <= 1.1e-62) {
		tmp = ((t - a) * z) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -4.2e-92)
		tmp = t_1;
	elseif (z <= 1e-125)
		tmp = fma(z, x, x);
	elseif (z <= 1.1e-62)
		tmp = Float64(Float64(Float64(t - a) * z) / y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.2e-92], t$95$1, If[LessEqual[z, 1e-125], N[(z * x + x), $MachinePrecision], If[LessEqual[z, 1.1e-62], N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.2e-92 or 1.10000000000000009e-62 < z

   1. Initial program 53.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if -4.2e-92 < z < 1.00000000000000001e-125

   1. Initial program 85.7%

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

   3. Taylor expanded in 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 66.1

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 66.1%

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

   if 1.00000000000000001e-125 < z < 1.10000000000000009e-62

   1. Initial program 91.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower--.f64 68.6

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 63.7%

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

Alternative 9: 72.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4.6 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{y} + x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4.6e-20)
     t_1
     (if (<= z 1.8e-9) (fma (+ (/ (- t a) y) x) z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -4.6e-20) {
		tmp = t_1;
	} else if (z <= 1.8e-9) {
		tmp = fma((((t - a) / y) + x), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -4.6e-20)
		tmp = t_1;
	elseif (z <= 1.8e-9)
		tmp = fma(Float64(Float64(Float64(t - a) / y) + x), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.6e-20], t$95$1, If[LessEqual[z, 1.8e-9], N[(N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.5999999999999998e-20 or 1.8e-9 < z

   1. Initial program 47.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 77.8

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

   5. Applied rewrites 77.8%

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

   if -4.5999999999999998e-20 < z < 1.8e-9

   1. Initial program 87.0%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 56.3

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

   5. Applied rewrites 56.3%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 66.1

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

   8. Applied rewrites 66.1%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 74.7%

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

   12. Taylor expanded in z around 0

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

   14. Applied rewrites 71.0%

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

Alternative 10: 44.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{a - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -9.2e+67)
     t_1
     (if (<= z 7.8e+53)
       (/ x (- 1.0 z))
       (if (<= z 3.2e+157) (/ (- a t) y) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -9.2e+67) {
		tmp = t_1;
	} else if (z <= 7.8e+53) {
		tmp = x / (1.0 - z);
	} else if (z <= 3.2e+157) {
		tmp = (a - t) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / (b - y)
    if (z <= (-9.2d+67)) then
        tmp = t_1
    else if (z <= 7.8d+53) then
        tmp = x / (1.0d0 - z)
    else if (z <= 3.2d+157) then
        tmp = (a - t) / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -9.2e+67) {
		tmp = t_1;
	} else if (z <= 7.8e+53) {
		tmp = x / (1.0 - z);
	} else if (z <= 3.2e+157) {
		tmp = (a - t) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t / (b - y)
	tmp = 0
	if z <= -9.2e+67:
		tmp = t_1
	elif z <= 7.8e+53:
		tmp = x / (1.0 - z)
	elif z <= 3.2e+157:
		tmp = (a - t) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -9.2e+67)
		tmp = t_1;
	elseif (z <= 7.8e+53)
		tmp = Float64(x / Float64(1.0 - z));
	elseif (z <= 3.2e+157)
		tmp = Float64(Float64(a - t) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t / (b - y);
	tmp = 0.0;
	if (z <= -9.2e+67)
		tmp = t_1;
	elseif (z <= 7.8e+53)
		tmp = x / (1.0 - z);
	elseif (z <= 3.2e+157)
		tmp = (a - t) / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.2e+67], t$95$1, If[LessEqual[z, 7.8e+53], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.2e+157], N[(N[(a - t), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.1999999999999994e67 or 3.1999999999999999e157 < z

   1. Initial program 33.8%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 27.7

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

   5. Applied rewrites 27.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 51.0%

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

   if -9.1999999999999994e67 < z < 7.79999999999999952e53

   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. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 52.8

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

   5. Applied rewrites 52.8%

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

   if 7.79999999999999952e53 < z < 3.1999999999999999e157

   1. Initial program 64.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 46.3

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

   8. Applied rewrites 46.3%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 64.1%

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

   12. Taylor expanded in z around inf

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

   14. Applied rewrites 56.0%

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

Alternative 11: 64.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.02 \cdot 10^{-100}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -4.2e-92) t_1 (if (<= z 2.02e-100) (* 1.0 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 <= -4.2e-92) {
		tmp = t_1;
	} else if (z <= 2.02e-100) {
		tmp = 1.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-4.2d-92)) then
        tmp = t_1
    else if (z <= 2.02d-100) then
        tmp = 1.0d0 * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -4.2e-92) {
		tmp = t_1;
	} else if (z <= 2.02e-100) {
		tmp = 1.0 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -4.2e-92:
		tmp = t_1
	elif z <= 2.02e-100:
		tmp = 1.0 * 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 <= -4.2e-92)
		tmp = t_1;
	elseif (z <= 2.02e-100)
		tmp = Float64(1.0 * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -4.2e-92)
		tmp = t_1;
	elseif (z <= 2.02e-100)
		tmp = 1.0 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.2e-92], t$95$1, If[LessEqual[z, 2.02e-100], N[(1.0 * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.2e-92 or 2.02000000000000005e-100 < z

   1. Initial program 55.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 69.8

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

   5. Applied rewrites 69.8%

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

   if -4.2e-92 < z < 2.02000000000000005e-100

   1. Initial program 85.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 64.0

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 80.7%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 64.0%

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

Alternative 12: 53.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{-95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -1.45e-95) t_1 (if (<= y 2.6e-43) (/ (- 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 <= -1.45e-95) {
		tmp = t_1;
	} else if (y <= 2.6e-43) {
		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 <= (-1.45d-95)) then
        tmp = t_1
    else if (y <= 2.6d-43) 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 <= -1.45e-95) {
		tmp = t_1;
	} else if (y <= 2.6e-43) {
		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 <= -1.45e-95:
		tmp = t_1
	elif y <= 2.6e-43:
		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 <= -1.45e-95)
		tmp = t_1;
	elseif (y <= 2.6e-43)
		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 <= -1.45e-95)
		tmp = t_1;
	elseif (y <= 2.6e-43)
		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, -1.45e-95], t$95$1, If[LessEqual[y, 2.6e-43], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45000000000000001e-95 or 2.6e-43 < y

   1. Initial program 59.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 50.0

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

   5. Applied rewrites 50.0%

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

   if -1.45000000000000001e-95 < y < 2.6e-43

   1. Initial program 81.8%

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

   3. Taylor expanded in y around 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 62.1

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

   5. Applied rewrites 62.1%

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

Alternative 13: 44.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, x, x\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -2.1e-30) t_1 (if (<= z 8.6e-17) (fma (fma z x x) z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -2.1e-30) {
		tmp = t_1;
	} else if (z <= 8.6e-17) {
		tmp = fma(fma(z, x, x), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -2.1e-30)
		tmp = t_1;
	elseif (z <= 8.6e-17)
		tmp = fma(fma(z, x, x), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.1e-30], t$95$1, If[LessEqual[z, 8.6e-17], N[(N[(z * x + x), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1000000000000002e-30 or 8.60000000000000046e-17 < z

   1. Initial program 48.8%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 25.8

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 40.9%

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

   if -2.1000000000000002e-30 < z < 8.60000000000000046e-17

   1. Initial program 87.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 58.0

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

   5. Applied rewrites 58.0%

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

   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 58.0%

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

Alternative 14: 36.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, x, x\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.1e-30)
   (/ t b)
   (if (<= z 8.6e-17) (fma (fma z x x) z x) (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.1e-30) {
		tmp = t / b;
	} else if (z <= 8.6e-17) {
		tmp = fma(fma(z, x, x), z, x);
	} else {
		tmp = t / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2.1e-30)
		tmp = Float64(t / b);
	elseif (z <= 8.6e-17)
		tmp = fma(fma(z, x, x), z, x);
	else
		tmp = Float64(t / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2.1e-30], N[(t / b), $MachinePrecision], If[LessEqual[z, 8.6e-17], N[(N[(z * x + x), $MachinePrecision] * z + x), $MachinePrecision], N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1000000000000002e-30 or 8.60000000000000046e-17 < z

   1. Initial program 48.8%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 25.8

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 28.0%

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

   if -2.1000000000000002e-30 < z < 8.60000000000000046e-17

   1. Initial program 87.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 58.0

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

   5. Applied rewrites 58.0%

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

   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 58.0%

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

Alternative 15: 36.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-17}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.1e-30) (/ t b) (if (<= z 8.6e-17) (* 1.0 x) (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.1e-30) {
		tmp = t / b;
	} else if (z <= 8.6e-17) {
		tmp = 1.0 * x;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-2.1d-30)) then
        tmp = t / b
    else if (z <= 8.6d-17) then
        tmp = 1.0d0 * x
    else
        tmp = t / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.1e-30) {
		tmp = t / b;
	} else if (z <= 8.6e-17) {
		tmp = 1.0 * x;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -2.1e-30:
		tmp = t / b
	elif z <= 8.6e-17:
		tmp = 1.0 * x
	else:
		tmp = t / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2.1e-30)
		tmp = Float64(t / b);
	elseif (z <= 8.6e-17)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(t / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -2.1e-30)
		tmp = t / b;
	elseif (z <= 8.6e-17)
		tmp = 1.0 * x;
	else
		tmp = t / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2.1e-30], N[(t / b), $MachinePrecision], If[LessEqual[z, 8.6e-17], N[(1.0 * x), $MachinePrecision], N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1000000000000002e-30 or 8.60000000000000046e-17 < z

   1. Initial program 48.8%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 25.8

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 28.0%

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

   if -2.1000000000000002e-30 < z < 8.60000000000000046e-17

   1. Initial program 87.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 58.0

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

   5. Applied rewrites 58.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 82.5%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 58.0%

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

Alternative 16: 25.1% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.7%

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

3. Taylor expanded in 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 38.6

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

5. Applied rewrites 38.6%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 30.9%

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

Alternative 17: 25.0% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.7%

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

3. Taylor expanded in 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 38.6

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

5. Applied rewrites 38.6%

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

6. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

8. Applied rewrites 66.0%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 30.8%

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

Alternative 18: 3.7% 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 67.7%

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

3. Taylor expanded in 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 38.6

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

5. Applied rewrites 38.6%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 30.9%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 3.2%

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

Developer Target 1: 73.4% 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 2024277 
(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)))))