Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.3% → 96.3%

Time: 12.0s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ \mathbf{if}\;z \leq -8800000000000 \lor \neg \left(z \leq 135000000000\right):\\ \;\;\;\;\frac{\frac{y}{b - y} \cdot \left(x - t\_1\right)}{z} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t\_2}, t - a, \frac{y}{t\_2} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))) (t_2 (fma (- b y) z y)))
   (if (or (<= z -8800000000000.0) (not (<= z 135000000000.0)))
     (+ (/ (* (/ y (- b y)) (- x t_1)) z) t_1)
     (fma (/ z t_2) (- t a) (* (/ y t_2) x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double t_2 = fma((b - y), z, y);
	double tmp;
	if ((z <= -8800000000000.0) || !(z <= 135000000000.0)) {
		tmp = (((y / (b - y)) * (x - t_1)) / z) + t_1;
	} else {
		tmp = fma((z / t_2), (t - a), ((y / t_2) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = fma(Float64(b - y), z, y)
	tmp = 0.0
	if ((z <= -8800000000000.0) || !(z <= 135000000000.0))
		tmp = Float64(Float64(Float64(Float64(y / Float64(b - y)) * Float64(x - t_1)) / z) + t_1);
	else
		tmp = fma(Float64(z / t_2), Float64(t - a), Float64(Float64(y / t_2) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, If[Or[LessEqual[z, -8800000000000.0], N[Not[LessEqual[z, 135000000000.0]], $MachinePrecision]], N[(N[(N[(N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision] * N[(x - t$95$1), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(z / t$95$2), $MachinePrecision] * N[(t - a), $MachinePrecision] + N[(N[(y / t$95$2), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.8e12 or 1.35e11 < z

   1. Initial program 39.3%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 99.9%

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

   if -8.8e12 < z < 1.35e11

   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 t around 0

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

      1. associate-+r+ N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. neg-mul-1 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 97.4%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8800000000000 \lor \neg \left(z \leq 135000000000\right):\\ \;\;\;\;\frac{\frac{y}{b - y} \cdot \left(x - \frac{t - a}{b - y}\right)}{z} + \frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(b - y, z, y\right)}, t - a, \frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_2 := {\left(b - y\right)}^{-1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_2, t - a, \frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-295}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t\_2, t - a, \frac{x}{z} \cdot \frac{y}{b - y}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, t - a, {\left(1 - z\right)}^{-1} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_2 (pow (- b y) -1.0)))
   (if (<= t_1 (- INFINITY))
     (fma t_2 (- t a) (* (/ y (fma (- b y) z y)) x))
     (if (<= t_1 -1e-295)
       t_1
       (if (<= t_1 0.0)
         (fma t_2 (- t a) (* (/ x z) (/ y (- b y))))
         (if (<= t_1 5e+298)
           t_1
           (fma t_2 (- t a) (* (pow (- 1.0 z) -1.0) x))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_2 = pow((b - y), -1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(t_2, (t - a), ((y / fma((b - y), z, y)) * x));
	} else if (t_1 <= -1e-295) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = fma(t_2, (t - a), ((x / z) * (y / (b - y))));
	} else if (t_1 <= 5e+298) {
		tmp = t_1;
	} else {
		tmp = fma(t_2, (t - a), (pow((1.0 - z), -1.0) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_2 = Float64(b - y) ^ -1.0
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(t_2, Float64(t - a), Float64(Float64(y / fma(Float64(b - y), z, y)) * x));
	elseif (t_1 <= -1e-295)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = fma(t_2, Float64(t - a), Float64(Float64(x / z) * Float64(y / Float64(b - y))));
	elseif (t_1 <= 5e+298)
		tmp = t_1;
	else
		tmp = fma(t_2, Float64(t - a), Float64((Float64(1.0 - z) ^ -1.0) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(b - y), $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$2 * N[(t - a), $MachinePrecision] + N[(N[(y / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-295], t$95$1, If[LessEqual[t$95$1, 0.0], N[(t$95$2 * N[(t - a), $MachinePrecision] + N[(N[(x / z), $MachinePrecision] * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+298], t$95$1, N[(t$95$2 * N[(t - a), $MachinePrecision] + N[(N[Power[N[(1.0 - z), $MachinePrecision], -1.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 11.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. neg-mul-1 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.7%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000006e-295 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 5.0000000000000003e298

   1. Initial program 99.6%

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

   if -1.00000000000000006e-295 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 31.4%

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

   3. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. neg-mul-1 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 31.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 86.5%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 99.5%

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

   if 5.0000000000000003e298 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 12.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. neg-mul-1 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 45.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 88.1%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 94.0%

       \[\leadsto \mathsf{fma}\left(\frac{1}{b - y}, t - a, \frac{1}{1 - z} \cdot x\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({\left(b - y\right)}^{-1}, t - a, \frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-295}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\mathsf{fma}\left({\left(b - y\right)}^{-1}, t - a, \frac{x}{z} \cdot \frac{y}{b - y}\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(b - y\right)}^{-1}, t - a, {\left(1 - z\right)}^{-1} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_2 := {\left(b - y\right)}^{-1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t\_2, t - a, \frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-295}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, t - a, {\left(1 - z\right)}^{-1} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_2 (pow (- b y) -1.0)))
   (if (<= t_1 (- INFINITY))
     (fma t_2 (- t a) (* (/ y (fma (- b y) z y)) x))
     (if (<= t_1 -1e-295)
       t_1
       (if (<= t_1 0.0)
         (/ (- t a) (- b y))
         (if (<= t_1 5e+298)
           t_1
           (fma t_2 (- t a) (* (pow (- 1.0 z) -1.0) x))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_2 = pow((b - y), -1.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(t_2, (t - a), ((y / fma((b - y), z, y)) * x));
	} else if (t_1 <= -1e-295) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (t - a) / (b - y);
	} else if (t_1 <= 5e+298) {
		tmp = t_1;
	} else {
		tmp = fma(t_2, (t - a), (pow((1.0 - z), -1.0) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_2 = Float64(b - y) ^ -1.0
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(t_2, Float64(t - a), Float64(Float64(y / fma(Float64(b - y), z, y)) * x));
	elseif (t_1 <= -1e-295)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(t - a) / Float64(b - y));
	elseif (t_1 <= 5e+298)
		tmp = t_1;
	else
		tmp = fma(t_2, Float64(t - a), Float64((Float64(1.0 - z) ^ -1.0) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(b - y), $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(t$95$2 * N[(t - a), $MachinePrecision] + N[(N[(y / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1e-295], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+298], t$95$1, N[(t$95$2 * N[(t - a), $MachinePrecision] + N[(N[Power[N[(1.0 - z), $MachinePrecision], -1.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 11.6%

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

   3. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. neg-mul-1 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 99.7%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000006e-295 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 5.0000000000000003e298

   1. Initial program 99.6%

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

   if -1.00000000000000006e-295 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 31.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 86.9

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

   5. Applied rewrites 86.9%

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

   if 5.0000000000000003e298 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 12.0%

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

   3. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. neg-mul-1 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 45.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 88.1%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 94.0%

       \[\leadsto \mathsf{fma}\left(\frac{1}{b - y}, t - a, \frac{1}{1 - z} \cdot x\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({\left(b - y\right)}^{-1}, t - a, \frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-295}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(b - y\right)}^{-1}, t - a, {\left(1 - z\right)}^{-1} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_2 := \mathsf{fma}\left({\left(b - y\right)}^{-1}, t - a, {\left(1 - z\right)}^{-1} \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-295}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_2 (fma (pow (- b y) -1.0) (- t a) (* (pow (- 1.0 z) -1.0) x))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 -1e-295)
       t_1
       (if (<= t_1 0.0) (/ (- t a) (- b y)) (if (<= t_1 5e+298) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_2 = fma(pow((b - y), -1.0), (t - a), (pow((1.0 - z), -1.0) * x));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= -1e-295) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (t - a) / (b - y);
	} else if (t_1 <= 5e+298) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_2 = fma((Float64(b - y) ^ -1.0), Float64(t - a), Float64((Float64(1.0 - z) ^ -1.0) * x))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= -1e-295)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(t - a) / Float64(b - y));
	elseif (t_1 <= 5e+298)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(b - y), $MachinePrecision], -1.0], $MachinePrecision] * N[(t - a), $MachinePrecision] + N[(N[Power[N[(1.0 - z), $MachinePrecision], -1.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, -1e-295], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+298], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 5.0000000000000003e298 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 11.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. neg-mul-1 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 90.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 91.7%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.00000000000000006e-295 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 5.0000000000000003e298

   1. Initial program 99.6%

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

   if -1.00000000000000006e-295 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 31.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 86.9

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

   5. Applied rewrites 86.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left({\left(b - y\right)}^{-1}, t - a, {\left(1 - z\right)}^{-1} \cdot x\right)\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{-295}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(b - y\right)}^{-1}, t - a, {\left(1 - z\right)}^{-1} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ \mathbf{if}\;z \leq -2 \cdot 10^{+16} \lor \neg \left(z \leq 135000000000\right):\\ \;\;\;\;\mathsf{fma}\left({\left(b - y\right)}^{-1}, t - a, \frac{x}{z} \cdot \frac{y}{b - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t\_1}, t - a, \frac{y}{t\_1} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y)))
   (if (or (<= z -2e+16) (not (<= z 135000000000.0)))
     (fma (pow (- b y) -1.0) (- t a) (* (/ x z) (/ y (- b y))))
     (fma (/ z t_1) (- t a) (* (/ y t_1) x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double tmp;
	if ((z <= -2e+16) || !(z <= 135000000000.0)) {
		tmp = fma(pow((b - y), -1.0), (t - a), ((x / z) * (y / (b - y))));
	} else {
		tmp = fma((z / t_1), (t - a), ((y / t_1) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	tmp = 0.0
	if ((z <= -2e+16) || !(z <= 135000000000.0))
		tmp = fma((Float64(b - y) ^ -1.0), Float64(t - a), Float64(Float64(x / z) * Float64(y / Float64(b - y))));
	else
		tmp = fma(Float64(z / t_1), Float64(t - a), Float64(Float64(y / t_1) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, If[Or[LessEqual[z, -2e+16], N[Not[LessEqual[z, 135000000000.0]], $MachinePrecision]], N[(N[Power[N[(b - y), $MachinePrecision], -1.0], $MachinePrecision] * N[(t - a), $MachinePrecision] + N[(N[(x / z), $MachinePrecision] * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / t$95$1), $MachinePrecision] * N[(t - a), $MachinePrecision] + N[(N[(y / t$95$1), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e16 or 1.35e11 < z

   1. Initial program 38.7%

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

   3. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. neg-mul-1 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 91.3%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 99.6%

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

   if -2e16 < z < 1.35e11

   1. Initial program 87.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. neg-mul-1 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 97.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+16} \lor \neg \left(z \leq 135000000000\right):\\ \;\;\;\;\mathsf{fma}\left({\left(b - y\right)}^{-1}, t - a, \frac{x}{z} \cdot \frac{y}{b - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\mathsf{fma}\left(b - y, z, y\right)}, t - a, \frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -9000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-88}:\\ \;\;\;\;\left(t - a\right) \cdot \frac{z}{t\_1}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{y}{t\_1} \cdot x\\ \mathbf{elif}\;z \leq 68000000000:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -9000000000.0)
     t_2
     (if (<= z -1.4e-88)
       (* (- t a) (/ z t_1))
       (if (<= z 3.2e-130)
         (* (/ y t_1) x)
         (if (<= z 68000000000.0) (/ (* (- t a) z) t_1) t_2))))))

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) / (b - y);
	double tmp;
	if (z <= -9000000000.0) {
		tmp = t_2;
	} else if (z <= -1.4e-88) {
		tmp = (t - a) * (z / t_1);
	} else if (z <= 3.2e-130) {
		tmp = (y / t_1) * x;
	} else if (z <= 68000000000.0) {
		tmp = ((t - a) * z) / t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -9000000000.0)
		tmp = t_2;
	elseif (z <= -1.4e-88)
		tmp = Float64(Float64(t - a) * Float64(z / t_1));
	elseif (z <= 3.2e-130)
		tmp = Float64(Float64(y / t_1) * x);
	elseif (z <= 68000000000.0)
		tmp = Float64(Float64(Float64(t - a) * z) / t_1);
	else
		tmp = t_2;
	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[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9000000000.0], t$95$2, If[LessEqual[z, -1.4e-88], N[(N[(t - a), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.2e-130], N[(N[(y / t$95$1), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 68000000000.0], N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -9e9 or 6.8e10 < z

   1. Initial program 38.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if -9e9 < z < -1.39999999999999988e-88

   1. Initial program 95.3%

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

   3. 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 79.4

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

   5. Applied rewrites 79.4%

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

   if -1.39999999999999988e-88 < z < 3.2e-130

   1. Initial program 86.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 73.1

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

   5. Applied rewrites 73.1%

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

   if 3.2e-130 < z < 6.8e10

   1. Initial program 87.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. neg-mul-1 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9000000000:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-88}:\\ \;\;\;\;\left(t - a\right) \cdot \frac{z}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{elif}\;z \leq 68000000000:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{\left(t - a\right) \cdot z}{t\_1}\\ t_3 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{+15}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-88}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{y}{t\_1} \cdot x\\ \mathbf{elif}\;z \leq 68000000000:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y))
        (t_2 (/ (* (- t a) z) t_1))
        (t_3 (/ (- t a) (- b y))))
   (if (<= z -2.2e+15)
     t_3
     (if (<= z -1.4e-88)
       t_2
       (if (<= z 3.2e-130)
         (* (/ y t_1) x)
         (if (<= z 68000000000.0) t_2 t_3))))))

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) / t_1;
	double t_3 = (t - a) / (b - y);
	double tmp;
	if (z <= -2.2e+15) {
		tmp = t_3;
	} else if (z <= -1.4e-88) {
		tmp = t_2;
	} else if (z <= 3.2e-130) {
		tmp = (y / t_1) * x;
	} else if (z <= 68000000000.0) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(Float64(t - a) * z) / t_1)
	t_3 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -2.2e+15)
		tmp = t_3;
	elseif (z <= -1.4e-88)
		tmp = t_2;
	elseif (z <= 3.2e-130)
		tmp = Float64(Float64(y / t_1) * x);
	elseif (z <= 68000000000.0)
		tmp = t_2;
	else
		tmp = t_3;
	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[(t - a), $MachinePrecision] * z), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.2e+15], t$95$3, If[LessEqual[z, -1.4e-88], t$95$2, If[LessEqual[z, 3.2e-130], N[(N[(y / t$95$1), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 68000000000.0], t$95$2, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2e15 or 6.8e10 < z

   1. Initial program 38.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 85.6

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

   5. Applied rewrites 85.6%

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

   if -2.2e15 < z < -1.39999999999999988e-88 or 3.2e-130 < z < 6.8e10

   1. Initial program 89.2%

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

   3. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. neg-mul-1 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.0%

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

   if -1.39999999999999988e-88 < z < 3.2e-130

   1. Initial program 86.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 73.1

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

   5. Applied rewrites 73.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-88}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{elif}\;z \leq 68000000000:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{+62} \lor \neg \left(z \leq 4.1 \cdot 10^{+93}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.35e+62) (not (<= z 4.1e+93)))
   (/ (- t a) (- b y))
   (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.35e+62) || !(z <= 4.1e+93)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	}
	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 <= (-2.35d+62)) .or. (.not. (z <= 4.1d+93))) then
        tmp = (t - a) / (b - y)
    else
        tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)))
    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 <= -2.35e+62) || !(z <= 4.1e+93)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -2.35e+62) or not (z <= 4.1e+93):
		tmp = (t - a) / (b - y)
	else:
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.35e+62) || !(z <= 4.1e+93))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -2.35e+62) || ~((z <= 4.1e+93)))
		tmp = (t - a) / (b - y);
	else
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.35e+62], N[Not[LessEqual[z, 4.1e+93]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3500000000000001e62 or 4.1000000000000001e93 < z

   1. Initial program 29.4%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 87.4

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

   5. Applied rewrites 87.4%

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

   if -2.3500000000000001e62 < z < 4.1000000000000001e93

   1. Initial program 86.9%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{+62} \lor \neg \left(z \leq 4.1 \cdot 10^{+93}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-47} \lor \neg \left(z \leq 1.75\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, t \cdot z\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -6.8e-47) (not (<= z 1.75)))
   (/ (- t a) (- b y))
   (/ (fma x y (* t z)) (fma (- b y) z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -6.8e-47) || !(z <= 1.75)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(x, y, (t * z)) / fma((b - y), z, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -6.8e-47) || !(z <= 1.75))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(fma(x, y, Float64(t * z)) / fma(Float64(b - y), z, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -6.8e-47], N[Not[LessEqual[z, 1.75]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(x * y + N[(t * z), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.8000000000000003e-47 or 1.75 < z

   1. Initial program 44.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 82.3

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

   5. Applied rewrites 82.3%

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

   if -6.8000000000000003e-47 < z < 1.75

   1. Initial program 87.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-+r+ N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. neg-mul-1 N/A

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

      8. sub-neg N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 68.7

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

   8. Applied rewrites 68.7%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-47} \lor \neg \left(z \leq 1.75\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, t \cdot z\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-47} \lor \neg \left(z \leq 1.75\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -6.8e-47) (not (<= z 1.75)))
   (/ (- t a) (- b y))
   (/ (fma t z (* y x)) (fma (- b y) z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -6.8e-47) || !(z <= 1.75)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(t, z, (y * x)) / fma((b - y), z, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -6.8e-47) || !(z <= 1.75))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(fma(t, z, Float64(y * x)) / fma(Float64(b - y), z, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -6.8e-47], N[Not[LessEqual[z, 1.75]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.8000000000000003e-47 or 1.75 < z

   1. Initial program 44.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 82.3

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

   5. Applied rewrites 82.3%

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

   if -6.8000000000000003e-47 < z < 1.75

   1. Initial program 87.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 68.7

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

   5. Applied rewrites 68.7%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-47} \lor \neg \left(z \leq 1.75\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-86} \lor \neg \left(z \leq 1.06 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.3e-86) (not (<= z 1.06e-10)))
   (/ (- t a) (- b y))
   (* (/ y (fma (- b y) z y)) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.3e-86) || !(z <= 1.06e-10)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = (y / fma((b - y), z, y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.3e-86) || !(z <= 1.06e-10))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(Float64(y / fma(Float64(b - y), z, y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.3e-86], N[Not[LessEqual[z, 1.06e-10]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.29999999999999996e-86 or 1.06e-10 < z

   1. Initial program 50.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 77.3

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

   5. Applied rewrites 77.3%

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

   if -2.29999999999999996e-86 < z < 1.06e-10

   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 x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 65.9

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

   5. Applied rewrites 65.9%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-86} \lor \neg \left(z \leq 1.06 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+162}:\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -3.4e-17)
     t_1
     (if (<= z 7e-34) (fma x z x) (if (<= z 1.02e+162) (/ (- a) b) 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 <= -3.4e-17) {
		tmp = t_1;
	} else if (z <= 7e-34) {
		tmp = fma(x, z, x);
	} else if (z <= 1.02e+162) {
		tmp = -a / b;
	} 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 <= -3.4e-17)
		tmp = t_1;
	elseif (z <= 7e-34)
		tmp = fma(x, z, x);
	elseif (z <= 1.02e+162)
		tmp = Float64(Float64(-a) / b);
	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, -3.4e-17], t$95$1, If[LessEqual[z, 7e-34], N[(x * z + x), $MachinePrecision], If[LessEqual[z, 1.02e+162], N[((-a) / b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.3999999999999998e-17 or 1.01999999999999993e162 < z

   1. Initial program 39.4%

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

   3. Taylor expanded in 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 29.9

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

   5. Applied rewrites 29.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 51.5%

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

   if -3.3999999999999998e-17 < z < 7e-34

   1. Initial program 86.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 55.9

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

   5. Applied rewrites 55.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 55.9%

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

   if 7e-34 < z < 1.01999999999999993e162

   1. Initial program 63.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 49.1

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

   5. Applied rewrites 49.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 42.6%

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

Alternative 13: 65.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-88} \lor \neg \left(z \leq 7 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -8.2e-88) (not (<= z 7e-34)))
   (/ (- t a) (- b y))
   (/ x (- 1.0 z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -8.2e-88) || !(z <= 7e-34)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = x / (1.0 - z);
	}
	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 <= (-8.2d-88)) .or. (.not. (z <= 7d-34))) then
        tmp = (t - a) / (b - y)
    else
        tmp = x / (1.0d0 - z)
    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 <= -8.2e-88) || !(z <= 7e-34)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = x / (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -8.2e-88) or not (z <= 7e-34):
		tmp = (t - a) / (b - y)
	else:
		tmp = x / (1.0 - z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -8.2e-88) || !(z <= 7e-34))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(x / Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -8.2e-88) || ~((z <= 7e-34)))
		tmp = (t - a) / (b - y);
	else
		tmp = x / (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -8.2e-88], N[Not[LessEqual[z, 7e-34]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.2000000000000002e-88 or 7e-34 < z

   1. Initial program 52.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 75.7

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

   5. Applied rewrites 75.7%

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

   if -8.2000000000000002e-88 < z < 7e-34

   1. Initial program 85.8%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 60.1

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

   5. Applied rewrites 60.1%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-88} \lor \neg \left(z \leq 7 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7000000000000 \lor \neg \left(y \leq 1.95 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -7000000000000.0) (not (<= y 1.95e+15)))
   (/ x (- 1.0 z))
   (/ (- t a) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -7000000000000.0) || !(y <= 1.95e+15)) {
		tmp = x / (1.0 - z);
	} else {
		tmp = (t - a) / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-7000000000000.0d0)) .or. (.not. (y <= 1.95d+15))) then
        tmp = x / (1.0d0 - z)
    else
        tmp = (t - a) / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -7000000000000.0) || !(y <= 1.95e+15)) {
		tmp = x / (1.0 - z);
	} else {
		tmp = (t - a) / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -7000000000000.0) or not (y <= 1.95e+15):
		tmp = x / (1.0 - z)
	else:
		tmp = (t - a) / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -7000000000000.0) || !(y <= 1.95e+15))
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = Float64(Float64(t - a) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -7000000000000.0) || ~((y <= 1.95e+15)))
		tmp = x / (1.0 - z);
	else
		tmp = (t - a) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -7000000000000.0], N[Not[LessEqual[y, 1.95e+15]], $MachinePrecision]], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7e12 or 1.95e15 < y

   1. Initial program 58.9%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 58.4

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

   5. Applied rewrites 58.4%

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

   if -7e12 < y < 1.95e15

   1. Initial program 76.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 55.4

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

   5. Applied rewrites 55.4%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7000000000000 \lor \neg \left(y \leq 1.95 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2800000000:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2800000000.0) (/ t b) (if (<= z 7e-34) (fma x z x) (/ (- a) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2800000000.0) {
		tmp = t / b;
	} else if (z <= 7e-34) {
		tmp = fma(x, z, x);
	} else {
		tmp = -a / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2800000000.0)
		tmp = Float64(t / b);
	elseif (z <= 7e-34)
		tmp = fma(x, z, x);
	else
		tmp = Float64(Float64(-a) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2800000000.0], N[(t / b), $MachinePrecision], If[LessEqual[z, 7e-34], N[(x * z + x), $MachinePrecision], N[((-a) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.8e9

   1. Initial program 38.1%

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

   3. Taylor expanded in 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 29.0

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

   5. Applied rewrites 29.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 35.2%

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

   if -2.8e9 < z < 7e-34

   1. Initial program 87.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 53.5

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

   5. Applied rewrites 53.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 53.7%

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

   if 7e-34 < z

   1. Initial program 51.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 51.5

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

   5. Applied rewrites 51.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 38.0%

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

Alternative 16: 36.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2800000000 \lor \neg \left(z \leq 10^{-14}\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2800000000.0) (not (<= z 1e-14))) (/ t b) (fma x z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2800000000.0) || !(z <= 1e-14)) {
		tmp = t / b;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2800000000.0) || !(z <= 1e-14))
		tmp = Float64(t / b);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2800000000.0], N[Not[LessEqual[z, 1e-14]], $MachinePrecision]], N[(t / b), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8e9 or 9.99999999999999999e-15 < z

   1. Initial program 43.2%

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

   3. Taylor expanded in 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 28.6

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

   5. Applied rewrites 28.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 28.2%

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

   if -2.8e9 < z < 9.99999999999999999e-15

   1. Initial program 87.1%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 53.3%

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

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2800000000 \lor \neg \left(z \leq 10^{-14}\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 25.6% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.4%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
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 37.2

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

5. Applied rewrites 37.2%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 30.8%

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

Alternative 18: 3.9% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.4%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
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 37.2

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

5. Applied rewrites 37.2%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 30.8%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 3.4%

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

Developer Target 1: 73.7% 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 2024313 
(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)))))