Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.0% → 86.8%

Time: 21.7s

Alternatives: 15

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+17}:\\ \;\;\;\;\frac{x \cdot \frac{y}{b - y} + y \cdot \frac{a - t}{{\left(b - y\right)}^{2}}}{z} + t\_1\\ \mathbf{elif}\;z \leq 550000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.1e+17)
     (+ (/ (+ (* x (/ y (- b y))) (* y (/ (- a t) (pow (- b y) 2.0)))) z) t_1)
     (if (<= z 550000000.0)
       (/ (fma x y (* z (- t a))) (+ y (* z (- b y))))
       (- t_1 (/ x z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.1e+17) {
		tmp = (((x * (y / (b - y))) + (y * ((a - t) / pow((b - y), 2.0)))) / z) + t_1;
	} else if (z <= 550000000.0) {
		tmp = fma(x, y, (z * (t - a))) / (y + (z * (b - y)));
	} else {
		tmp = t_1 - (x / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.1e+17)
		tmp = Float64(Float64(Float64(Float64(x * Float64(y / Float64(b - y))) + Float64(y * Float64(Float64(a - t) / (Float64(b - y) ^ 2.0)))) / z) + t_1);
	elseif (z <= 550000000.0)
		tmp = Float64(fma(x, y, Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))));
	else
		tmp = Float64(t_1 - Float64(x / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.1e+17], N[(N[(N[(N[(x * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * N[(N[(a - t), $MachinePrecision] / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[z, 550000000.0], N[(N[(x * y + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - N[(x / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1e17

   1. Initial program 40.8%

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

      1. div-inv 40.7%

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

      2. fma-define 40.7%

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

      3. +-commutative 40.7%

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

      4. fma-define 40.7%

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

   4. Applied egg-rr 40.7%

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

   5. Taylor expanded in z around -inf 66.0%

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

      1. associate--l+ 66.0%

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

      2. mul-1-neg 66.0%

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

      3. distribute-lft-out-- 66.0%

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

      4. associate-/l* 76.5%

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

      5. associate-/l* 90.7%

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

      6. div-sub 90.7%

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

   7. Simplified 90.7%

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

   if -1.1e17 < z < 5.5e8

   1. Initial program 89.9%

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

      1. fma-define 89.9%

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

   3. Simplified 89.9%

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

   if 5.5e8 < z

   1. Initial program 36.2%

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

      1. div-inv 36.1%

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

      2. fma-define 36.1%

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

      3. +-commutative 36.1%

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

      4. fma-define 36.1%

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

   4. Applied egg-rr 36.1%

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

   5. Taylor expanded in z around -inf 69.1%

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

      1. associate--l+ 69.1%

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

      2. mul-1-neg 69.1%

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

      3. distribute-lft-out-- 69.1%

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

      4. associate-/l* 74.1%

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

      5. associate-/l* 92.5%

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

      6. div-sub 92.5%

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

   7. Simplified 92.5%

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

   8. Taylor expanded in y around inf 93.9%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+17}:\\ \;\;\;\;\frac{x \cdot \frac{y}{b - y} + y \cdot \frac{a - t}{{\left(b - y\right)}^{2}}}{z} + \frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 550000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y} - \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -2.15 \cdot 10^{+15}:\\ \;\;\;\;\left(\left(\frac{t}{b - y} + \frac{y}{b - y} \cdot \frac{x}{z}\right) + \frac{a}{y - b}\right) + \frac{y}{z} \cdot \frac{t\_1}{y - b}\\ \mathbf{elif}\;z \leq 550000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -2.15e+15)
     (+
      (+ (+ (/ t (- b y)) (* (/ y (- b y)) (/ x z))) (/ a (- y b)))
      (* (/ y z) (/ t_1 (- y b))))
     (if (<= z 550000000.0)
       (/ (fma x y (* z (- t a))) (+ y (* z (- b y))))
       (- t_1 (/ x z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -2.15e+15) {
		tmp = (((t / (b - y)) + ((y / (b - y)) * (x / z))) + (a / (y - b))) + ((y / z) * (t_1 / (y - b)));
	} else if (z <= 550000000.0) {
		tmp = fma(x, y, (z * (t - a))) / (y + (z * (b - y)));
	} else {
		tmp = t_1 - (x / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -2.15e+15)
		tmp = Float64(Float64(Float64(Float64(t / Float64(b - y)) + Float64(Float64(y / Float64(b - y)) * Float64(x / z))) + Float64(a / Float64(y - b))) + Float64(Float64(y / z) * Float64(t_1 / Float64(y - b))));
	elseif (z <= 550000000.0)
		tmp = Float64(fma(x, y, Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))));
	else
		tmp = Float64(t_1 - Float64(x / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.15e+15], N[(N[(N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / N[(y - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y / z), $MachinePrecision] * N[(t$95$1 / N[(y - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 550000000.0], N[(N[(x * y + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - N[(x / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.15e15

   1. Initial program 40.8%

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

   3. Taylor expanded in z around inf 65.9%

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

      1. associate--r+ 65.9%

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

      2. associate-/r* 69.1%

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

      3. *-commutative 69.1%

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

      4. times-frac 75.5%

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

   5. Simplified 75.5%

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

      1. *-un-lft-identity 75.5%

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

      2. unpow2 75.5%

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

      3. times-frac 75.5%

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

   7. Applied egg-rr 75.5%

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

      1. associate-*l/ 75.5%

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

      2. *-lft-identity 75.5%

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

   9. Simplified 75.5%

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

       1. associate-/l/ 72.2%

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

       2. times-frac 89.1%

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

   11. Applied egg-rr 89.1%

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

   if -2.15e15 < z < 5.5e8

   1. Initial program 89.9%

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

      1. fma-define 89.9%

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

   3. Simplified 89.9%

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

   if 5.5e8 < z

   1. Initial program 36.2%

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

      1. div-inv 36.1%

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

      2. fma-define 36.1%

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

      3. +-commutative 36.1%

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

      4. fma-define 36.1%

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

   4. Applied egg-rr 36.1%

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

   5. Taylor expanded in z around -inf 69.1%

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

      1. associate--l+ 69.1%

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

      2. mul-1-neg 69.1%

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

      3. distribute-lft-out-- 69.1%

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

      4. associate-/l* 74.1%

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

      5. associate-/l* 92.5%

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

      6. div-sub 92.5%

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

   7. Simplified 92.5%

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

   8. Taylor expanded in y around inf 93.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+15}:\\ \;\;\;\;\left(\left(\frac{t}{b - y} + \frac{y}{b - y} \cdot \frac{x}{z}\right) + \frac{a}{y - b}\right) + \frac{y}{z} \cdot \frac{\frac{t - a}{b - y}}{y - b}\\ \mathbf{elif}\;z \leq 550000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y} - \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -9 \cdot 10^{+19}:\\ \;\;\;\;\left(\left(\frac{t}{b - y} + \frac{y}{b - y} \cdot \frac{x}{z}\right) + \frac{a}{y - b}\right) + \frac{y}{z} \cdot \frac{t\_1}{y - b}\\ \mathbf{elif}\;z \leq 550000000:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -9e+19)
     (+
      (+ (+ (/ t (- b y)) (* (/ y (- b y)) (/ x z))) (/ a (- y b)))
      (* (/ y z) (/ t_1 (- y b))))
     (if (<= z 550000000.0)
       (/ (+ (* z (- t a)) (* x y)) (+ y (* z (- b y))))
       (- t_1 (/ x z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -9e+19) {
		tmp = (((t / (b - y)) + ((y / (b - y)) * (x / z))) + (a / (y - b))) + ((y / z) * (t_1 / (y - b)));
	} else if (z <= 550000000.0) {
		tmp = ((z * (t - a)) + (x * y)) / (y + (z * (b - y)));
	} else {
		tmp = t_1 - (x / 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) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-9d+19)) then
        tmp = (((t / (b - y)) + ((y / (b - y)) * (x / z))) + (a / (y - b))) + ((y / z) * (t_1 / (y - b)))
    else if (z <= 550000000.0d0) then
        tmp = ((z * (t - a)) + (x * y)) / (y + (z * (b - y)))
    else
        tmp = t_1 - (x / 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 t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -9e+19) {
		tmp = (((t / (b - y)) + ((y / (b - y)) * (x / z))) + (a / (y - b))) + ((y / z) * (t_1 / (y - b)));
	} else if (z <= 550000000.0) {
		tmp = ((z * (t - a)) + (x * y)) / (y + (z * (b - y)));
	} else {
		tmp = t_1 - (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -9e+19:
		tmp = (((t / (b - y)) + ((y / (b - y)) * (x / z))) + (a / (y - b))) + ((y / z) * (t_1 / (y - b)))
	elif z <= 550000000.0:
		tmp = ((z * (t - a)) + (x * y)) / (y + (z * (b - y)))
	else:
		tmp = t_1 - (x / z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -9e+19)
		tmp = Float64(Float64(Float64(Float64(t / Float64(b - y)) + Float64(Float64(y / Float64(b - y)) * Float64(x / z))) + Float64(a / Float64(y - b))) + Float64(Float64(y / z) * Float64(t_1 / Float64(y - b))));
	elseif (z <= 550000000.0)
		tmp = Float64(Float64(Float64(z * Float64(t - a)) + Float64(x * y)) / Float64(y + Float64(z * Float64(b - y))));
	else
		tmp = Float64(t_1 - Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -9e+19)
		tmp = (((t / (b - y)) + ((y / (b - y)) * (x / z))) + (a / (y - b))) + ((y / z) * (t_1 / (y - b)));
	elseif (z <= 550000000.0)
		tmp = ((z * (t - a)) + (x * y)) / (y + (z * (b - y)));
	else
		tmp = t_1 - (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9e+19], N[(N[(N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / N[(y - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y / z), $MachinePrecision] * N[(t$95$1 / N[(y - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 550000000.0], N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - N[(x / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9e19

   1. Initial program 40.8%

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

   3. Taylor expanded in z around inf 65.9%

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

      1. associate--r+ 65.9%

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

      2. associate-/r* 69.1%

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

      3. *-commutative 69.1%

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

      4. times-frac 75.5%

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

   5. Simplified 75.5%

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

      1. *-un-lft-identity 75.5%

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

      2. unpow2 75.5%

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

      3. times-frac 75.5%

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

   7. Applied egg-rr 75.5%

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

      1. associate-*l/ 75.5%

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

      2. *-lft-identity 75.5%

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

   9. Simplified 75.5%

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

       1. associate-/l/ 72.2%

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

       2. times-frac 89.1%

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

   11. Applied egg-rr 89.1%

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

   if -9e19 < z < 5.5e8

   1. Initial program 89.9%

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

   if 5.5e8 < z

   1. Initial program 36.2%

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

      1. div-inv 36.1%

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

      2. fma-define 36.1%

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

      3. +-commutative 36.1%

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

      4. fma-define 36.1%

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

   4. Applied egg-rr 36.1%

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

   5. Taylor expanded in z around -inf 69.1%

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

      1. associate--l+ 69.1%

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

      2. mul-1-neg 69.1%

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

      3. distribute-lft-out-- 69.1%

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

      4. associate-/l* 74.1%

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

      5. associate-/l* 92.5%

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

      6. div-sub 92.5%

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

   7. Simplified 92.5%

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

   8. Taylor expanded in y around inf 93.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+19}:\\ \;\;\;\;\left(\left(\frac{t}{b - y} + \frac{y}{b - y} \cdot \frac{x}{z}\right) + \frac{a}{y - b}\right) + \frac{y}{z} \cdot \frac{\frac{t - a}{b - y}}{y - b}\\ \mathbf{elif}\;z \leq 550000000:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y} - \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y} - \frac{x}{z}\\ \mathbf{if}\;z \leq -1 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-140}:\\ \;\;\;\;\frac{-1}{\frac{z \cdot \left(y - b\right) - y}{z \cdot \left(t - a\right)}}\\ \mathbf{elif}\;z \leq 115000:\\ \;\;\;\;\frac{1}{\frac{y + z \cdot b}{x \cdot y + z \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (/ (- t a) (- b y)) (/ x z))))
   (if (<= z -1e+24)
     t_1
     (if (<= z -1e-140)
       (/ -1.0 (/ (- (* z (- y b)) y) (* z (- t a))))
       (if (<= z 115000.0)
         (/ 1.0 (/ (+ y (* z b)) (+ (* x y) (* z t))))
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((t - a) / (b - y)) - (x / z);
	double tmp;
	if (z <= -1e+24) {
		tmp = t_1;
	} else if (z <= -1e-140) {
		tmp = -1.0 / (((z * (y - b)) - y) / (z * (t - a)));
	} else if (z <= 115000.0) {
		tmp = 1.0 / ((y + (z * b)) / ((x * y) + (z * t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((t - a) / (b - y)) - (x / z)
    if (z <= (-1d+24)) then
        tmp = t_1
    else if (z <= (-1d-140)) then
        tmp = (-1.0d0) / (((z * (y - b)) - y) / (z * (t - a)))
    else if (z <= 115000.0d0) then
        tmp = 1.0d0 / ((y + (z * b)) / ((x * y) + (z * t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((t - a) / (b - y)) - (x / z);
	double tmp;
	if (z <= -1e+24) {
		tmp = t_1;
	} else if (z <= -1e-140) {
		tmp = -1.0 / (((z * (y - b)) - y) / (z * (t - a)));
	} else if (z <= 115000.0) {
		tmp = 1.0 / ((y + (z * b)) / ((x * y) + (z * t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = ((t - a) / (b - y)) - (x / z)
	tmp = 0
	if z <= -1e+24:
		tmp = t_1
	elif z <= -1e-140:
		tmp = -1.0 / (((z * (y - b)) - y) / (z * (t - a)))
	elif z <= 115000.0:
		tmp = 1.0 / ((y + (z * b)) / ((x * y) + (z * t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(t - a) / Float64(b - y)) - Float64(x / z))
	tmp = 0.0
	if (z <= -1e+24)
		tmp = t_1;
	elseif (z <= -1e-140)
		tmp = Float64(-1.0 / Float64(Float64(Float64(z * Float64(y - b)) - y) / Float64(z * Float64(t - a))));
	elseif (z <= 115000.0)
		tmp = Float64(1.0 / Float64(Float64(y + Float64(z * b)) / Float64(Float64(x * y) + Float64(z * t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((t - a) / (b - y)) - (x / z);
	tmp = 0.0;
	if (z <= -1e+24)
		tmp = t_1;
	elseif (z <= -1e-140)
		tmp = -1.0 / (((z * (y - b)) - y) / (z * (t - a)));
	elseif (z <= 115000.0)
		tmp = 1.0 / ((y + (z * b)) / ((x * y) + (z * t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1e+24], t$95$1, If[LessEqual[z, -1e-140], N[(-1.0 / N[(N[(N[(z * N[(y - b), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] / N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 115000.0], N[(1.0 / N[(N[(y + N[(z * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.9999999999999998e23 or 115000 < z

   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. Step-by-step derivation

      1. div-inv 38.0%

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

      2. fma-define 38.0%

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

      3. +-commutative 38.0%

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

      4. fma-define 38.0%

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

   4. Applied egg-rr 38.0%

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

   5. Taylor expanded in z around -inf 67.2%

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

      1. associate--l+ 67.2%

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

      2. mul-1-neg 67.2%

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

      3. distribute-lft-out-- 67.2%

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

      4. associate-/l* 75.2%

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

      5. associate-/l* 91.5%

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

      6. div-sub 91.5%

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

   7. Simplified 91.5%

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

   8. Taylor expanded in y around inf 87.6%

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

   if -9.9999999999999998e23 < z < -9.9999999999999998e-141

   1. Initial program 96.5%

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

      1. clear-num 96.2%

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

      2. inv-pow 96.2%

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

      3. +-commutative 96.2%

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

      4. fma-define 96.2%

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

      5. fma-define 96.2%

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

   4. Applied egg-rr 96.2%

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

      1. unpow-1 96.2%

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

      2. fma-define 96.2%

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

      3. +-commutative 96.2%

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

      4. fma-define 96.2%

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

      5. *-commutative 96.2%

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

   6. Simplified 96.2%

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

   7. Taylor expanded in x around 0 79.8%

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

   if -9.9999999999999998e-141 < z < 115000

   1. Initial program 88.1%

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

      1. clear-num 87.9%

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

      2. inv-pow 87.9%

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

      3. +-commutative 87.9%

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

      4. fma-define 87.9%

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

      5. fma-define 87.9%

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

   4. Applied egg-rr 87.9%

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

      1. unpow-1 87.9%

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

      2. fma-define 87.9%

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

      3. +-commutative 87.9%

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

      4. fma-define 87.9%

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

      5. *-commutative 87.9%

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

   6. Simplified 87.9%

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

   7. Taylor expanded in a around 0 73.7%

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

   8. Taylor expanded in b around inf 73.7%

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

      1. *-commutative 73.7%

         \[\leadsto \frac{1}{\frac{y + \color{blue}{z \cdot b}}{t \cdot z + x \cdot y}} \]

   10. Simplified 73.7%

       \[\leadsto \frac{1}{\frac{y + \color{blue}{z \cdot b}}{t \cdot z + x \cdot y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+24}:\\ \;\;\;\;\frac{t - a}{b - y} - \frac{x}{z}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-140}:\\ \;\;\;\;\frac{-1}{\frac{z \cdot \left(y - b\right) - y}{z \cdot \left(t - a\right)}}\\ \mathbf{elif}\;z \leq 115000:\\ \;\;\;\;\frac{1}{\frac{y + z \cdot b}{x \cdot y + z \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y} - \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+28} \lor \neg \left(z \leq 170000000\right):\\ \;\;\;\;\frac{t - a}{b - y} - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -6.2e+28) (not (<= z 170000000.0)))
   (- (/ (- t a) (- b y)) (/ x z))
   (/ (+ (* z (- t a)) (* x y)) (+ y (* z (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -6.2e+28) || !(z <= 170000000.0)) {
		tmp = ((t - a) / (b - y)) - (x / z);
	} else {
		tmp = ((z * (t - a)) + (x * y)) / (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 <= (-6.2d+28)) .or. (.not. (z <= 170000000.0d0))) then
        tmp = ((t - a) / (b - y)) - (x / z)
    else
        tmp = ((z * (t - a)) + (x * y)) / (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 <= -6.2e+28) || !(z <= 170000000.0)) {
		tmp = ((t - a) / (b - y)) - (x / z);
	} else {
		tmp = ((z * (t - a)) + (x * y)) / (y + (z * (b - y)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -6.2e+28) or not (z <= 170000000.0):
		tmp = ((t - a) / (b - y)) - (x / z)
	else:
		tmp = ((z * (t - a)) + (x * y)) / (y + (z * (b - y)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -6.2e+28) || !(z <= 170000000.0))
		tmp = Float64(Float64(Float64(t - a) / Float64(b - y)) - Float64(x / z));
	else
		tmp = Float64(Float64(Float64(z * Float64(t - a)) + Float64(x * y)) / 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 <= -6.2e+28) || ~((z <= 170000000.0)))
		tmp = ((t - a) / (b - y)) - (x / z);
	else
		tmp = ((z * (t - a)) + (x * y)) / (y + (z * (b - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -6.2e+28], N[Not[LessEqual[z, 170000000.0]], $MachinePrecision]], N[(N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.2000000000000001e28 or 1.7e8 < z

   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. Step-by-step derivation

      1. div-inv 38.0%

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

      2. fma-define 38.0%

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

      3. +-commutative 38.0%

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

      4. fma-define 38.0%

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

   4. Applied egg-rr 38.0%

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

   5. Taylor expanded in z around -inf 67.2%

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

      1. associate--l+ 67.2%

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

      2. mul-1-neg 67.2%

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

      3. distribute-lft-out-- 67.2%

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

      4. associate-/l* 75.2%

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

      5. associate-/l* 91.5%

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

      6. div-sub 91.5%

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

   7. Simplified 91.5%

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

   8. Taylor expanded in y around inf 87.6%

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

   if -6.2000000000000001e28 < z < 1.7e8

   1. Initial program 89.9%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+28} \lor \neg \left(z \leq 170000000\right):\\ \;\;\;\;\frac{t - a}{b - y} - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y + z \cdot \left(b - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y} - \frac{x}{z}\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{\left(t + \frac{x \cdot y}{z}\right) - a}{b}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (/ (- t a) (- b y)) (/ x z))))
   (if (<= z -8.2e+23)
     t_1
     (if (<= z -1.2e-96)
       (/ (- (+ t (/ (* x y) z)) a) b)
       (if (<= z 3.8e-20) (/ (+ (* z (- t a)) (* x y)) y) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((t - a) / (b - y)) - (x / z);
	double tmp;
	if (z <= -8.2e+23) {
		tmp = t_1;
	} else if (z <= -1.2e-96) {
		tmp = ((t + ((x * y) / z)) - a) / b;
	} else if (z <= 3.8e-20) {
		tmp = ((z * (t - a)) + (x * y)) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((t - a) / (b - y)) - (x / z)
    if (z <= (-8.2d+23)) then
        tmp = t_1
    else if (z <= (-1.2d-96)) then
        tmp = ((t + ((x * y) / z)) - a) / b
    else if (z <= 3.8d-20) then
        tmp = ((z * (t - a)) + (x * y)) / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((t - a) / (b - y)) - (x / z);
	double tmp;
	if (z <= -8.2e+23) {
		tmp = t_1;
	} else if (z <= -1.2e-96) {
		tmp = ((t + ((x * y) / z)) - a) / b;
	} else if (z <= 3.8e-20) {
		tmp = ((z * (t - a)) + (x * y)) / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = ((t - a) / (b - y)) - (x / z)
	tmp = 0
	if z <= -8.2e+23:
		tmp = t_1
	elif z <= -1.2e-96:
		tmp = ((t + ((x * y) / z)) - a) / b
	elif z <= 3.8e-20:
		tmp = ((z * (t - a)) + (x * y)) / y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(t - a) / Float64(b - y)) - Float64(x / z))
	tmp = 0.0
	if (z <= -8.2e+23)
		tmp = t_1;
	elseif (z <= -1.2e-96)
		tmp = Float64(Float64(Float64(t + Float64(Float64(x * y) / z)) - a) / b);
	elseif (z <= 3.8e-20)
		tmp = Float64(Float64(Float64(z * Float64(t - a)) + Float64(x * y)) / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((t - a) / (b - y)) - (x / z);
	tmp = 0.0;
	if (z <= -8.2e+23)
		tmp = t_1;
	elseif (z <= -1.2e-96)
		tmp = ((t + ((x * y) / z)) - a) / b;
	elseif (z <= 3.8e-20)
		tmp = ((z * (t - a)) + (x * y)) / y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.2e+23], t$95$1, If[LessEqual[z, -1.2e-96], N[(N[(N[(t + N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[z, 3.8e-20], N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.19999999999999992e23 or 3.7999999999999998e-20 < z

   1. Initial program 40.5%

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

      1. div-inv 40.5%

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

      2. fma-define 40.5%

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

      3. +-commutative 40.5%

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

      4. fma-define 40.5%

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

   4. Applied egg-rr 40.5%

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

   5. Taylor expanded in z around -inf 68.5%

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

      1. associate--l+ 68.5%

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

      2. mul-1-neg 68.5%

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

      3. distribute-lft-out-- 68.5%

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

      4. associate-/l* 75.4%

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

      5. associate-/l* 91.0%

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

      6. div-sub 91.0%

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

   7. Simplified 91.0%

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

   8. Taylor expanded in y around inf 86.3%

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

   if -8.19999999999999992e23 < z < -1.2000000000000001e-96

   1. Initial program 95.4%

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

   3. Taylor expanded in z around inf 64.7%

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

      1. associate--r+ 64.7%

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

      2. associate-/r* 64.7%

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

      3. *-commutative 64.7%

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

      4. times-frac 61.1%

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

   5. Simplified 61.1%

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

      1. *-un-lft-identity 61.1%

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

      2. unpow2 61.1%

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

      3. times-frac 61.1%

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

   7. Applied egg-rr 61.1%

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

      1. associate-*l/ 61.1%

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

      2. *-lft-identity 61.1%

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

   9. Simplified 61.1%

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

   10. Taylor expanded in b around inf 78.3%

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

   if -1.2000000000000001e-96 < z < 3.7999999999999998e-20

   1. Initial program 88.4%

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

   3. Taylor expanded in z around 0 65.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{t - a}{b - y} - \frac{x}{z}\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{\left(t + \frac{x \cdot y}{z}\right) - a}{b}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y} - \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.23 \lor \neg \left(z \leq 36000\right):\\ \;\;\;\;\frac{t - a}{b - y} - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y + z \cdot b}{x \cdot y + z \cdot t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -0.23) (not (<= z 36000.0)))
   (- (/ (- t a) (- b y)) (/ x z))
   (/ 1.0 (/ (+ y (* z b)) (+ (* x y) (* z t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -0.23) || !(z <= 36000.0)) {
		tmp = ((t - a) / (b - y)) - (x / z);
	} else {
		tmp = 1.0 / ((y + (z * b)) / ((x * y) + (z * t)));
	}
	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 <= (-0.23d0)) .or. (.not. (z <= 36000.0d0))) then
        tmp = ((t - a) / (b - y)) - (x / z)
    else
        tmp = 1.0d0 / ((y + (z * b)) / ((x * y) + (z * t)))
    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 <= -0.23) || !(z <= 36000.0)) {
		tmp = ((t - a) / (b - y)) - (x / z);
	} else {
		tmp = 1.0 / ((y + (z * b)) / ((x * y) + (z * t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -0.23) or not (z <= 36000.0):
		tmp = ((t - a) / (b - y)) - (x / z)
	else:
		tmp = 1.0 / ((y + (z * b)) / ((x * y) + (z * t)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -0.23) || !(z <= 36000.0))
		tmp = Float64(Float64(Float64(t - a) / Float64(b - y)) - Float64(x / z));
	else
		tmp = Float64(1.0 / Float64(Float64(y + Float64(z * b)) / Float64(Float64(x * y) + Float64(z * t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -0.23) || ~((z <= 36000.0)))
		tmp = ((t - a) / (b - y)) - (x / z);
	else
		tmp = 1.0 / ((y + (z * b)) / ((x * y) + (z * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -0.23], N[Not[LessEqual[z, 36000.0]], $MachinePrecision]], N[(N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(x / z), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y + N[(z * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.23000000000000001 or 36000 < z

   1. Initial program 39.1%

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

      1. div-inv 39.1%

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

      2. fma-define 39.1%

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

      3. +-commutative 39.1%

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

      4. fma-define 39.1%

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

   4. Applied egg-rr 39.1%

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

   5. Taylor expanded in z around -inf 67.8%

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

      1. associate--l+ 67.8%

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

      2. mul-1-neg 67.8%

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

      3. distribute-lft-out-- 67.8%

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

      4. associate-/l* 75.6%

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

      5. associate-/l* 91.6%

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

      6. div-sub 91.6%

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

   7. Simplified 91.6%

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

   8. Taylor expanded in y around inf 87.0%

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

   if -0.23000000000000001 < z < 36000

   1. Initial program 89.8%

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

      1. clear-num 89.6%

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

      2. inv-pow 89.6%

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

      3. +-commutative 89.6%

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

      4. fma-define 89.6%

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

      5. fma-define 89.6%

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

   4. Applied egg-rr 89.6%

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

      1. unpow-1 89.6%

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

      2. fma-define 89.6%

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

      3. +-commutative 89.6%

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

      4. fma-define 89.6%

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

      5. *-commutative 89.6%

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

   6. Simplified 89.6%

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

   7. Taylor expanded in a around 0 71.7%

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

   8. Taylor expanded in b around inf 71.7%

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

      1. *-commutative 71.7%

         \[\leadsto \frac{1}{\frac{y + \color{blue}{z \cdot b}}{t \cdot z + x \cdot y}} \]

   10. Simplified 71.7%

       \[\leadsto \frac{1}{\frac{y + \color{blue}{z \cdot b}}{t \cdot z + x \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.23 \lor \neg \left(z \leq 36000\right):\\ \;\;\;\;\frac{t - a}{b - y} - \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y + z \cdot b}{x \cdot y + z \cdot t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{-b}\\ \mathbf{if}\;z \leq -6.6 \cdot 10^{+240}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-140} \lor \neg \left(z \leq 2.8 \cdot 10^{-18}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (- b))))
   (if (<= z -6.6e+240)
     t_1
     (if (<= z -3.5e-45)
       (/ t b)
       (if (or (<= z -1e-140) (not (<= z 2.8e-18))) t_1 x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / -b;
	double tmp;
	if (z <= -6.6e+240) {
		tmp = t_1;
	} else if (z <= -3.5e-45) {
		tmp = t / b;
	} else if ((z <= -1e-140) || !(z <= 2.8e-18)) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a / -b
    if (z <= (-6.6d+240)) then
        tmp = t_1
    else if (z <= (-3.5d-45)) then
        tmp = t / b
    else if ((z <= (-1d-140)) .or. (.not. (z <= 2.8d-18))) then
        tmp = t_1
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / -b;
	double tmp;
	if (z <= -6.6e+240) {
		tmp = t_1;
	} else if (z <= -3.5e-45) {
		tmp = t / b;
	} else if ((z <= -1e-140) || !(z <= 2.8e-18)) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / -b
	tmp = 0
	if z <= -6.6e+240:
		tmp = t_1
	elif z <= -3.5e-45:
		tmp = t / b
	elif (z <= -1e-140) or not (z <= 2.8e-18):
		tmp = t_1
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(-b))
	tmp = 0.0
	if (z <= -6.6e+240)
		tmp = t_1;
	elseif (z <= -3.5e-45)
		tmp = Float64(t / b);
	elseif ((z <= -1e-140) || !(z <= 2.8e-18))
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / -b;
	tmp = 0.0;
	if (z <= -6.6e+240)
		tmp = t_1;
	elseif (z <= -3.5e-45)
		tmp = t / b;
	elseif ((z <= -1e-140) || ~((z <= 2.8e-18)))
		tmp = t_1;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / (-b)), $MachinePrecision]}, If[LessEqual[z, -6.6e+240], t$95$1, If[LessEqual[z, -3.5e-45], N[(t / b), $MachinePrecision], If[Or[LessEqual[z, -1e-140], N[Not[LessEqual[z, 2.8e-18]], $MachinePrecision]], t$95$1, x]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.5999999999999997e240 or -3.5e-45 < z < -9.9999999999999998e-141 or 2.80000000000000012e-18 < z

   1. Initial program 48.0%

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

   3. Taylor expanded in y around 0 51.4%

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

   4. Taylor expanded in t around 0 35.7%

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

      1. associate-*r/ 35.7%

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

      2. mul-1-neg 35.7%

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

   6. Simplified 35.7%

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

   if -6.5999999999999997e240 < z < -3.5e-45

   1. Initial program 55.9%

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

   3. Taylor expanded in t around inf 36.3%

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

      1. *-commutative 36.3%

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

   5. Simplified 36.3%

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

   6. Taylor expanded in y around 0 41.8%

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

   if -9.9999999999999998e-141 < z < 2.80000000000000012e-18

   1. Initial program 87.6%

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

   3. Taylor expanded in z around 0 60.2%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+240}:\\ \;\;\;\;\frac{a}{-b}\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-140} \lor \neg \left(z \leq 2.8 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{a}{-b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-98} \lor \neg \left(z \leq 3.8 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.7e-98) (not (<= z 3.8e-16)))
   (/ (- t a) (- b y))
   (/ (+ (* z (- t a)) (* x y)) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.7e-98) || !(z <= 3.8e-16)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = ((z * (t - a)) + (x * y)) / 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 <= (-1.7d-98)) .or. (.not. (z <= 3.8d-16))) then
        tmp = (t - a) / (b - y)
    else
        tmp = ((z * (t - a)) + (x * y)) / 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 <= -1.7e-98) || !(z <= 3.8e-16)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = ((z * (t - a)) + (x * y)) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -1.7e-98) or not (z <= 3.8e-16):
		tmp = (t - a) / (b - y)
	else:
		tmp = ((z * (t - a)) + (x * y)) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.7e-98) || !(z <= 3.8e-16))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(Float64(Float64(z * Float64(t - a)) + Float64(x * y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -1.7e-98) || ~((z <= 3.8e-16)))
		tmp = (t - a) / (b - y);
	else
		tmp = ((z * (t - a)) + (x * y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.7e-98], N[Not[LessEqual[z, 3.8e-16]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.7000000000000001e-98 or 3.80000000000000012e-16 < z

   1. Initial program 48.7%

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

   3. Taylor expanded in z around inf 76.4%

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

   if -1.7000000000000001e-98 < z < 3.80000000000000012e-16

   1. Initial program 88.4%

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

   3. Taylor expanded in z around 0 65.2%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-98} \lor \neg \left(z \leq 3.8 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + x \cdot y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-140}:\\ \;\;\;\;\frac{a}{-b}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -2.8e-45)
     t_1
     (if (<= z -1e-140) (/ a (- b)) (if (<= z 3.1e-5) x t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -2.8e-45) {
		tmp = t_1;
	} else if (z <= -1e-140) {
		tmp = a / -b;
	} else if (z <= 3.1e-5) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / (b - y)
    if (z <= (-2.8d-45)) then
        tmp = t_1
    else if (z <= (-1d-140)) then
        tmp = a / -b
    else if (z <= 3.1d-5) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -2.8e-45) {
		tmp = t_1;
	} else if (z <= -1e-140) {
		tmp = a / -b;
	} else if (z <= 3.1e-5) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t / (b - y)
	tmp = 0
	if z <= -2.8e-45:
		tmp = t_1
	elif z <= -1e-140:
		tmp = a / -b
	elif z <= 3.1e-5:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -2.8e-45)
		tmp = t_1;
	elseif (z <= -1e-140)
		tmp = Float64(a / Float64(-b));
	elseif (z <= 3.1e-5)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t / (b - y);
	tmp = 0.0;
	if (z <= -2.8e-45)
		tmp = t_1;
	elseif (z <= -1e-140)
		tmp = a / -b;
	elseif (z <= 3.1e-5)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.8e-45], t$95$1, If[LessEqual[z, -1e-140], N[(a / (-b)), $MachinePrecision], If[LessEqual[z, 3.1e-5], x, t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.8000000000000001e-45 or 3.10000000000000014e-5 < z

   1. Initial program 44.6%

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

   3. Taylor expanded in t around inf 26.4%

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

      1. *-commutative 26.4%

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

   5. Simplified 26.4%

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

   6. Taylor expanded in z around inf 44.5%

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

   if -2.8000000000000001e-45 < z < -9.9999999999999998e-141

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0 49.8%

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

   4. Taylor expanded in t around 0 44.1%

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

      1. associate-*r/ 44.1%

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

      2. mul-1-neg 44.1%

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

   6. Simplified 44.1%

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

   if -9.9999999999999998e-141 < z < 3.10000000000000014e-5

   1. Initial program 87.7%

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

   3. Taylor expanded in z around 0 59.7%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{-45}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-140}:\\ \;\;\;\;\frac{a}{-b}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{-45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-140}:\\ \;\;\;\;\frac{a}{-b}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -2.5e-45)
     t_1
     (if (<= z -1e-140) (/ a (- b)) (if (<= z 4.6e+48) (/ x (- 1.0 z)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -2.5e-45) {
		tmp = t_1;
	} else if (z <= -1e-140) {
		tmp = a / -b;
	} else if (z <= 4.6e+48) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / (b - y)
    if (z <= (-2.5d-45)) then
        tmp = t_1
    else if (z <= (-1d-140)) then
        tmp = a / -b
    else if (z <= 4.6d+48) then
        tmp = x / (1.0d0 - z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -2.5e-45) {
		tmp = t_1;
	} else if (z <= -1e-140) {
		tmp = a / -b;
	} else if (z <= 4.6e+48) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t / (b - y)
	tmp = 0
	if z <= -2.5e-45:
		tmp = t_1
	elif z <= -1e-140:
		tmp = a / -b
	elif z <= 4.6e+48:
		tmp = x / (1.0 - z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -2.5e-45)
		tmp = t_1;
	elseif (z <= -1e-140)
		tmp = Float64(a / Float64(-b));
	elseif (z <= 4.6e+48)
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t / (b - y);
	tmp = 0.0;
	if (z <= -2.5e-45)
		tmp = t_1;
	elseif (z <= -1e-140)
		tmp = a / -b;
	elseif (z <= 4.6e+48)
		tmp = x / (1.0 - z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.5e-45], t$95$1, If[LessEqual[z, -1e-140], N[(a / (-b)), $MachinePrecision], If[LessEqual[z, 4.6e+48], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.49999999999999988e-45 or 4.6e48 < z

   1. Initial program 41.8%

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

   3. Taylor expanded in t around inf 25.3%

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

      1. *-commutative 25.3%

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

   5. Simplified 25.3%

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

   6. Taylor expanded in z around inf 45.7%

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

   if -2.49999999999999988e-45 < z < -9.9999999999999998e-141

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0 49.8%

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

   4. Taylor expanded in t around 0 44.1%

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

      1. associate-*r/ 44.1%

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

      2. mul-1-neg 44.1%

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

   6. Simplified 44.1%

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

   if -9.9999999999999998e-141 < z < 4.6e48

   1. Initial program 85.1%

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

   3. Taylor expanded in y around inf 57.5%

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

      1. mul-1-neg 57.5%

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

      2. unsub-neg 57.5%

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

   5. Simplified 57.5%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-140}:\\ \;\;\;\;\frac{a}{-b}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-153} \lor \neg \left(z \leq 1.15 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.5e-153) (not (<= z 1.15e-50))) (/ (- t a) (- b y)) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.5e-153) || !(z <= 1.15e-50)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = x;
	}
	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 <= (-2.5d-153)) .or. (.not. (z <= 1.15d-50))) then
        tmp = (t - a) / (b - y)
    else
        tmp = x
    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 <= -2.5e-153) || !(z <= 1.15e-50)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -2.5e-153) or not (z <= 1.15e-50):
		tmp = (t - a) / (b - y)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.5e-153) || !(z <= 1.15e-50))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -2.5e-153) || ~((z <= 1.15e-50)))
		tmp = (t - a) / (b - y);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.5e-153], N[Not[LessEqual[z, 1.15e-50]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -2.50000000000000016e-153 or 1.1500000000000001e-50 < z

   1. Initial program 54.7%

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

   3. Taylor expanded in z around inf 73.0%

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

   if -2.50000000000000016e-153 < z < 1.1500000000000001e-50

   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 z around 0 64.2%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-153} \lor \neg \left(z \leq 1.15 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-10} \lor \neg \left(y \leq 4.7 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -7.2e-10) (not (<= y 4.7e+45))) (/ 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 <= -7.2e-10) || !(y <= 4.7e+45)) {
		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 <= (-7.2d-10)) .or. (.not. (y <= 4.7d+45))) 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 <= -7.2e-10) || !(y <= 4.7e+45)) {
		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 <= -7.2e-10) or not (y <= 4.7e+45):
		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 <= -7.2e-10) || !(y <= 4.7e+45))
		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 <= -7.2e-10) || ~((y <= 4.7e+45)))
		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, -7.2e-10], N[Not[LessEqual[y, 4.7e+45]], $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 < -7.2e-10 or 4.70000000000000002e45 < y

   1. Initial program 56.2%

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

   3. Taylor expanded in y around inf 57.4%

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

      1. mul-1-neg 57.4%

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

      2. unsub-neg 57.4%

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

   5. Simplified 57.4%

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

   if -7.2e-10 < y < 4.70000000000000002e45

   1. Initial program 73.3%

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

   3. Taylor expanded in y around 0 63.5%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-10} \lor \neg \left(y \leq 4.7 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{-64} \lor \neg \left(z \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -4.9e-64) (not (<= z 2e-8))) (/ t b) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -4.9e-64) || !(z <= 2e-8)) {
		tmp = t / b;
	} else {
		tmp = x;
	}
	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 <= (-4.9d-64)) .or. (.not. (z <= 2d-8))) then
        tmp = t / b
    else
        tmp = x
    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 <= -4.9e-64) || !(z <= 2e-8)) {
		tmp = t / b;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -4.9e-64) or not (z <= 2e-8):
		tmp = t / b
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -4.9e-64) || !(z <= 2e-8))
		tmp = Float64(t / b);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -4.9e-64) || ~((z <= 2e-8)))
		tmp = t / b;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -4.9e-64], N[Not[LessEqual[z, 2e-8]], $MachinePrecision]], N[(t / b), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -4.9000000000000002e-64 or 2e-8 < z

   1. Initial program 46.2%

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

   3. Taylor expanded in t around inf 26.4%

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

      1. *-commutative 26.4%

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

   5. Simplified 26.4%

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

   6. Taylor expanded in y around 0 30.0%

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

   if -4.9000000000000002e-64 < z < 2e-8

   1. Initial program 89.1%

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

   3. Taylor expanded in z around 0 55.1%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{-64} \lor \neg \left(z \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 25.3% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.4%

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

3. Taylor expanded in z around 0 27.3%

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

4. Final simplification 27.3%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 73.7% accurate, 0.7× 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 2024046 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :alt
  (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z))))

  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))