Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.4% → 90.8%

Time: 21.7s

Alternatives: 24

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+259}:\\ \;\;\;\;\frac{z}{y} \cdot \frac{a - t}{z + -1} - \frac{x}{z + -1}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{\frac{y \cdot x}{b - y} + \frac{a - t}{\frac{{\left(b - y\right)}^{2}}{y}}}{z} + \frac{t - a}{b - y}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+283}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \left(\frac{x}{z} + \frac{a}{b - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* z (- t a)) (* y x)) (+ y (* z (- b y))))))
   (if (<= t_1 -2e+259)
     (- (* (/ z y) (/ (- a t) (+ z -1.0))) (/ x (+ z -1.0)))
     (if (<= t_1 -2e-308)
       t_1
       (if (<= t_1 0.0)
         (+
          (/ (+ (/ (* y x) (- b y)) (/ (- a t) (/ (pow (- b y) 2.0) y))) z)
          (/ (- t a) (- b y)))
         (if (<= t_1 5e+283)
           t_1
           (- (/ t (- b y)) (+ (/ x z) (/ a (- b y))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((z * (t - a)) + (y * x)) / (y + (z * (b - y)));
	double tmp;
	if (t_1 <= -2e+259) {
		tmp = ((z / y) * ((a - t) / (z + -1.0))) - (x / (z + -1.0));
	} else if (t_1 <= -2e-308) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = ((((y * x) / (b - y)) + ((a - t) / (pow((b - y), 2.0) / y))) / z) + ((t - a) / (b - y));
	} else if (t_1 <= 5e+283) {
		tmp = t_1;
	} else {
		tmp = (t / (b - y)) - ((x / z) + (a / (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) :: t_1
    real(8) :: tmp
    t_1 = ((z * (t - a)) + (y * x)) / (y + (z * (b - y)))
    if (t_1 <= (-2d+259)) then
        tmp = ((z / y) * ((a - t) / (z + (-1.0d0)))) - (x / (z + (-1.0d0)))
    else if (t_1 <= (-2d-308)) then
        tmp = t_1
    else if (t_1 <= 0.0d0) then
        tmp = ((((y * x) / (b - y)) + ((a - t) / (((b - y) ** 2.0d0) / y))) / z) + ((t - a) / (b - y))
    else if (t_1 <= 5d+283) then
        tmp = t_1
    else
        tmp = (t / (b - y)) - ((x / z) + (a / (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 t_1 = ((z * (t - a)) + (y * x)) / (y + (z * (b - y)));
	double tmp;
	if (t_1 <= -2e+259) {
		tmp = ((z / y) * ((a - t) / (z + -1.0))) - (x / (z + -1.0));
	} else if (t_1 <= -2e-308) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = ((((y * x) / (b - y)) + ((a - t) / (Math.pow((b - y), 2.0) / y))) / z) + ((t - a) / (b - y));
	} else if (t_1 <= 5e+283) {
		tmp = t_1;
	} else {
		tmp = (t / (b - y)) - ((x / z) + (a / (b - y)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = ((z * (t - a)) + (y * x)) / (y + (z * (b - y)))
	tmp = 0
	if t_1 <= -2e+259:
		tmp = ((z / y) * ((a - t) / (z + -1.0))) - (x / (z + -1.0))
	elif t_1 <= -2e-308:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = ((((y * x) / (b - y)) + ((a - t) / (math.pow((b - y), 2.0) / y))) / z) + ((t - a) / (b - y))
	elif t_1 <= 5e+283:
		tmp = t_1
	else:
		tmp = (t / (b - y)) - ((x / z) + (a / (b - y)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(z * Float64(t - a)) + Float64(y * x)) / Float64(y + Float64(z * Float64(b - y))))
	tmp = 0.0
	if (t_1 <= -2e+259)
		tmp = Float64(Float64(Float64(z / y) * Float64(Float64(a - t) / Float64(z + -1.0))) - Float64(x / Float64(z + -1.0)));
	elseif (t_1 <= -2e-308)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(Float64(y * x) / Float64(b - y)) + Float64(Float64(a - t) / Float64((Float64(b - y) ^ 2.0) / y))) / z) + Float64(Float64(t - a) / Float64(b - y)));
	elseif (t_1 <= 5e+283)
		tmp = t_1;
	else
		tmp = Float64(Float64(t / Float64(b - y)) - Float64(Float64(x / z) + Float64(a / Float64(b - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((z * (t - a)) + (y * x)) / (y + (z * (b - y)));
	tmp = 0.0;
	if (t_1 <= -2e+259)
		tmp = ((z / y) * ((a - t) / (z + -1.0))) - (x / (z + -1.0));
	elseif (t_1 <= -2e-308)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = ((((y * x) / (b - y)) + ((a - t) / (((b - y) ^ 2.0) / y))) / z) + ((t - a) / (b - y));
	elseif (t_1 <= 5e+283)
		tmp = t_1;
	else
		tmp = (t / (b - y)) - ((x / z) + (a / (b - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+259], N[(N[(N[(z / y), $MachinePrecision] * N[(N[(a - t), $MachinePrecision] / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e-308], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(N[(N[(N[(y * x), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(N[(a - t), $MachinePrecision] / N[(N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+283], t$95$1, N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(N[(x / z), $MachinePrecision] + N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -2e259

   1. Initial program 35.6%

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

   2. Taylor expanded in y around -inf 16.3%

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

      1. +-commutative 16.3%

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

      2. mul-1-neg 16.3%

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

      3. unsub-neg 16.3%

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

   4. Simplified 50.8%

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

   5. Taylor expanded in b around 0 60.4%

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

      1. times-frac 76.7%

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

      2. sub-neg 76.7%

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

      3. metadata-eval 76.7%

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

   7. Simplified 76.7%

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

   if -2e259 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.9999999999999998e-308 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 5.0000000000000004e283

   1. Initial program 99.6%

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

   if -1.9999999999999998e-308 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0

   1. Initial program 20.8%

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

   2. Taylor expanded in z around -inf 83.2%

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

      1. +-commutative 83.2%

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

      2. associate--l+ 83.2%

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

   4. Simplified 94.3%

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

   if 5.0000000000000004e283 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 6.6%

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

   2. Taylor expanded in z around -inf 38.2%

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

      1. associate--l+ 38.2%

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

      2. mul-1-neg 38.2%

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

      3. distribute-lft-out-- 38.2%

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

      4. associate-/l* 54.9%

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

      5. associate-/l* 89.3%

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

   4. Simplified 89.3%

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

   5. Taylor expanded in y around inf 88.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)} \leq -2 \cdot 10^{+259}:\\ \;\;\;\;\frac{z}{y} \cdot \frac{a - t}{z + -1} - \frac{x}{z + -1}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)} \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{\frac{y \cdot x}{b - y} + \frac{a - t}{\frac{{\left(b - y\right)}^{2}}{y}}}{z} + \frac{t - a}{b - y}\\ \mathbf{elif}\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)} \leq 5 \cdot 10^{+283}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \left(\frac{x}{z} + \frac{a}{b - y}\right)\\ \end{array} \]

Alternative 2: 91.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -250000000000 \lor \neg \left(z \leq 23000000\right):\\ \;\;\;\;\frac{y}{b - y} \cdot \frac{x}{z} - \left(\left(\frac{a}{b - y} - \frac{y}{{\left(b - y\right)}^{2}} \cdot \frac{a - t}{z}\right) - \frac{t}{b - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -250000000000.0) (not (<= z 23000000.0)))
   (-
    (* (/ y (- b y)) (/ x z))
    (-
     (- (/ a (- b y)) (* (/ y (pow (- b y) 2.0)) (/ (- a t) z)))
     (/ t (- b y))))
   (/ (fma x y (* z (- t a))) (fma z (- b y) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -250000000000.0) || !(z <= 23000000.0)) {
		tmp = ((y / (b - y)) * (x / z)) - (((a / (b - y)) - ((y / pow((b - y), 2.0)) * ((a - t) / z))) - (t / (b - y)));
	} else {
		tmp = fma(x, y, (z * (t - a))) / fma(z, (b - y), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -250000000000.0) || !(z <= 23000000.0))
		tmp = Float64(Float64(Float64(y / Float64(b - y)) * Float64(x / z)) - Float64(Float64(Float64(a / Float64(b - y)) - Float64(Float64(y / (Float64(b - y) ^ 2.0)) * Float64(Float64(a - t) / z))) - Float64(t / Float64(b - y))));
	else
		tmp = Float64(fma(x, y, Float64(z * Float64(t - a))) / fma(z, Float64(b - y), y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -250000000000.0], N[Not[LessEqual[z, 23000000.0]], $MachinePrecision]], N[(N[(N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(N[(y / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(a - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.5e11 or 2.3e7 < z

   1. Initial program 39.6%

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

   2. Taylor expanded in z around inf 62.3%

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

      1. associate--l+ 62.3%

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

      2. *-commutative 62.3%

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

      3. times-frac 76.8%

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

      4. +-commutative 76.8%

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

      5. times-frac 96.1%

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

   4. Simplified 96.1%

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

   if -2.5e11 < z < 2.3e7

   1. Initial program 89.4%

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

      1. fma-def 89.4%

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

      2. +-commutative 89.4%

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

      3. fma-def 89.4%

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

   3. Simplified 89.4%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -250000000000 \lor \neg \left(z \leq 23000000\right):\\ \;\;\;\;\frac{y}{b - y} \cdot \frac{x}{z} - \left(\left(\frac{a}{b - y} - \frac{y}{{\left(b - y\right)}^{2}} \cdot \frac{a - t}{z}\right) - \frac{t}{b - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, z \cdot \left(t - a\right)\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \end{array} \]

Alternative 3: 91.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3950000000 \lor \neg \left(z \leq 1350000\right):\\ \;\;\;\;\frac{y}{b - y} \cdot \frac{x}{z} - \left(\left(\frac{a}{b - y} - \frac{y}{{\left(b - y\right)}^{2}} \cdot \frac{a - t}{z}\right) - \frac{t}{b - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -3950000000.0) (not (<= z 1350000.0)))
   (-
    (* (/ y (- b y)) (/ x z))
    (-
     (- (/ a (- b y)) (* (/ y (pow (- b y) 2.0)) (/ (- a t) z)))
     (/ t (- b y))))
   (/ (+ (* z (- t a)) (* y x)) (+ y (* z (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -3950000000.0) || !(z <= 1350000.0)) {
		tmp = ((y / (b - y)) * (x / z)) - (((a / (b - y)) - ((y / pow((b - y), 2.0)) * ((a - t) / z))) - (t / (b - y)));
	} else {
		tmp = ((z * (t - a)) + (y * x)) / (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 <= (-3950000000.0d0)) .or. (.not. (z <= 1350000.0d0))) then
        tmp = ((y / (b - y)) * (x / z)) - (((a / (b - y)) - ((y / ((b - y) ** 2.0d0)) * ((a - t) / z))) - (t / (b - y)))
    else
        tmp = ((z * (t - a)) + (y * x)) / (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 <= -3950000000.0) || !(z <= 1350000.0)) {
		tmp = ((y / (b - y)) * (x / z)) - (((a / (b - y)) - ((y / Math.pow((b - y), 2.0)) * ((a - t) / z))) - (t / (b - y)));
	} else {
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * (b - y)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -3950000000.0) or not (z <= 1350000.0):
		tmp = ((y / (b - y)) * (x / z)) - (((a / (b - y)) - ((y / math.pow((b - y), 2.0)) * ((a - t) / z))) - (t / (b - y)))
	else:
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * (b - y)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -3950000000.0) || !(z <= 1350000.0))
		tmp = Float64(Float64(Float64(y / Float64(b - y)) * Float64(x / z)) - Float64(Float64(Float64(a / Float64(b - y)) - Float64(Float64(y / (Float64(b - y) ^ 2.0)) * Float64(Float64(a - t) / z))) - Float64(t / Float64(b - y))));
	else
		tmp = Float64(Float64(Float64(z * Float64(t - a)) + Float64(y * x)) / 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 <= -3950000000.0) || ~((z <= 1350000.0)))
		tmp = ((y / (b - y)) * (x / z)) - (((a / (b - y)) - ((y / ((b - y) ^ 2.0)) * ((a - t) / z))) - (t / (b - y)));
	else
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * (b - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -3950000000.0], N[Not[LessEqual[z, 1350000.0]], $MachinePrecision]], N[(N[(N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(N[(y / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(a - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.95e9 or 1.35e6 < z

   1. Initial program 39.6%

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

   2. Taylor expanded in z around inf 62.3%

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

      1. associate--l+ 62.3%

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

      2. *-commutative 62.3%

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

      3. times-frac 76.8%

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

      4. +-commutative 76.8%

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

      5. times-frac 96.1%

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

   4. Simplified 96.1%

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

   if -3.95e9 < z < 1.35e6

   1. Initial program 89.4%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3950000000 \lor \neg \left(z \leq 1350000\right):\\ \;\;\;\;\frac{y}{b - y} \cdot \frac{x}{z} - \left(\left(\frac{a}{b - y} - \frac{y}{{\left(b - y\right)}^{2}} \cdot \frac{a - t}{z}\right) - \frac{t}{b - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)}\\ \end{array} \]

Alternative 4: 71.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + z \cdot \left(x + \frac{t - a}{y}\right)\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-270}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-183}:\\ \;\;\;\;\frac{y \cdot x + z \cdot t}{y + z \cdot b}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.26 \cdot 10^{+16}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (* z (+ x (/ (- t a) y))))))
   (if (<= z -5.2e-6)
     (/ (- t a) (- b y))
     (if (<= z -1.3e-270)
       t_1
       (if (<= z 3.8e-183)
         (/ (+ (* y x) (* z t)) (+ y (* z b)))
         (if (<= z 1.2e-94)
           t_1
           (if (<= z 1.26e+16)
             (/ (* z (- t a)) (+ y (* z (- b y))))
             (- (/ t (- b y)) (/ a (- b y))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (z * (x + ((t - a) / y)));
	double tmp;
	if (z <= -5.2e-6) {
		tmp = (t - a) / (b - y);
	} else if (z <= -1.3e-270) {
		tmp = t_1;
	} else if (z <= 3.8e-183) {
		tmp = ((y * x) + (z * t)) / (y + (z * b));
	} else if (z <= 1.2e-94) {
		tmp = t_1;
	} else if (z <= 1.26e+16) {
		tmp = (z * (t - a)) / (y + (z * (b - y)));
	} else {
		tmp = (t / (b - y)) - (a / (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) :: t_1
    real(8) :: tmp
    t_1 = x + (z * (x + ((t - a) / y)))
    if (z <= (-5.2d-6)) then
        tmp = (t - a) / (b - y)
    else if (z <= (-1.3d-270)) then
        tmp = t_1
    else if (z <= 3.8d-183) then
        tmp = ((y * x) + (z * t)) / (y + (z * b))
    else if (z <= 1.2d-94) then
        tmp = t_1
    else if (z <= 1.26d+16) then
        tmp = (z * (t - a)) / (y + (z * (b - y)))
    else
        tmp = (t / (b - y)) - (a / (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 t_1 = x + (z * (x + ((t - a) / y)));
	double tmp;
	if (z <= -5.2e-6) {
		tmp = (t - a) / (b - y);
	} else if (z <= -1.3e-270) {
		tmp = t_1;
	} else if (z <= 3.8e-183) {
		tmp = ((y * x) + (z * t)) / (y + (z * b));
	} else if (z <= 1.2e-94) {
		tmp = t_1;
	} else if (z <= 1.26e+16) {
		tmp = (z * (t - a)) / (y + (z * (b - y)));
	} else {
		tmp = (t / (b - y)) - (a / (b - y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x + (z * (x + ((t - a) / y)))
	tmp = 0
	if z <= -5.2e-6:
		tmp = (t - a) / (b - y)
	elif z <= -1.3e-270:
		tmp = t_1
	elif z <= 3.8e-183:
		tmp = ((y * x) + (z * t)) / (y + (z * b))
	elif z <= 1.2e-94:
		tmp = t_1
	elif z <= 1.26e+16:
		tmp = (z * (t - a)) / (y + (z * (b - y)))
	else:
		tmp = (t / (b - y)) - (a / (b - y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(z * Float64(x + Float64(Float64(t - a) / y))))
	tmp = 0.0
	if (z <= -5.2e-6)
		tmp = Float64(Float64(t - a) / Float64(b - y));
	elseif (z <= -1.3e-270)
		tmp = t_1;
	elseif (z <= 3.8e-183)
		tmp = Float64(Float64(Float64(y * x) + Float64(z * t)) / Float64(y + Float64(z * b)));
	elseif (z <= 1.2e-94)
		tmp = t_1;
	elseif (z <= 1.26e+16)
		tmp = Float64(Float64(z * Float64(t - a)) / Float64(y + Float64(z * Float64(b - y))));
	else
		tmp = Float64(Float64(t / Float64(b - y)) - Float64(a / Float64(b - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x + (z * (x + ((t - a) / y)));
	tmp = 0.0;
	if (z <= -5.2e-6)
		tmp = (t - a) / (b - y);
	elseif (z <= -1.3e-270)
		tmp = t_1;
	elseif (z <= 3.8e-183)
		tmp = ((y * x) + (z * t)) / (y + (z * b));
	elseif (z <= 1.2e-94)
		tmp = t_1;
	elseif (z <= 1.26e+16)
		tmp = (z * (t - a)) / (y + (z * (b - y)));
	else
		tmp = (t / (b - y)) - (a / (b - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(z * N[(x + N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.2e-6], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.3e-270], t$95$1, If[LessEqual[z, 3.8e-183], N[(N[(N[(y * x), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.2e-94], t$95$1, If[LessEqual[z, 1.26e+16], N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -5.20000000000000019e-6

   1. Initial program 41.6%

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

   2. Taylor expanded in z around inf 77.7%

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

   if -5.20000000000000019e-6 < z < -1.3000000000000001e-270 or 3.7999999999999996e-183 < z < 1.2e-94

   1. Initial program 88.1%

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

   2. Taylor expanded in y around -inf 59.0%

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

      1. +-commutative 59.0%

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

      2. mul-1-neg 59.0%

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

      3. unsub-neg 59.0%

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

   4. Simplified 74.1%

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

   5. Taylor expanded in b around 0 81.2%

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

      1. times-frac 74.8%

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

      2. sub-neg 74.8%

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

      3. metadata-eval 74.8%

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

   7. Simplified 74.8%

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

   8. Taylor expanded in z around 0 75.7%

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

   if -1.3000000000000001e-270 < z < 3.7999999999999996e-183

   1. Initial program 91.2%

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

   2. Taylor expanded in b around inf 91.2%

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

   3. Taylor expanded in a around 0 81.1%

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

   if 1.2e-94 < z < 1.26e16

   1. Initial program 90.4%

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

      1. fma-def 90.4%

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

      2. +-commutative 90.4%

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

      3. fma-def 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in x around 0 71.1%

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

   if 1.26e16 < z

   1. Initial program 38.1%

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

   2. Taylor expanded in z around -inf 70.3%

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

      1. associate--l+ 70.3%

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

      2. mul-1-neg 70.3%

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

      3. distribute-lft-out-- 70.3%

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

      4. associate-/l* 73.6%

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

      5. associate-/l* 91.3%

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

   4. Simplified 91.3%

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

   5. Taylor expanded in y around inf 87.9%

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

   6. Taylor expanded in x around 0 72.4%

      \[\leadsto \color{blue}{\frac{t}{b - y} - \frac{a}{b - y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-270}:\\ \;\;\;\;x + z \cdot \left(x + \frac{t - a}{y}\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-183}:\\ \;\;\;\;\frac{y \cdot x + z \cdot t}{y + z \cdot b}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-94}:\\ \;\;\;\;x + z \cdot \left(x + \frac{t - a}{y}\right)\\ \mathbf{elif}\;z \leq 1.26 \cdot 10^{+16}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \end{array} \]

Alternative 5: 62.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot x - z \cdot a}{y}\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{-164}:\\ \;\;\;\;x \cdot \frac{1}{1 - z}\\ \mathbf{elif}\;z \leq -2.65 \cdot 10^{-275}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-183}:\\ \;\;\;\;\frac{y \cdot x}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-79}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (* y x) (* z a)) y)))
   (if (<= z -7.5e-6)
     (/ (- t a) (- b y))
     (if (<= z -9.6e-164)
       (* x (/ 1.0 (- 1.0 z)))
       (if (<= z -2.65e-275)
         t_1
         (if (<= z 8.8e-183)
           (/ (* y x) (+ y (* z (- b y))))
           (if (<= z 4.4e-79) t_1 (- (/ t (- b y)) (/ a (- b y))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y * x) - (z * a)) / y;
	double tmp;
	if (z <= -7.5e-6) {
		tmp = (t - a) / (b - y);
	} else if (z <= -9.6e-164) {
		tmp = x * (1.0 / (1.0 - z));
	} else if (z <= -2.65e-275) {
		tmp = t_1;
	} else if (z <= 8.8e-183) {
		tmp = (y * x) / (y + (z * (b - y)));
	} else if (z <= 4.4e-79) {
		tmp = t_1;
	} else {
		tmp = (t / (b - y)) - (a / (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) :: t_1
    real(8) :: tmp
    t_1 = ((y * x) - (z * a)) / y
    if (z <= (-7.5d-6)) then
        tmp = (t - a) / (b - y)
    else if (z <= (-9.6d-164)) then
        tmp = x * (1.0d0 / (1.0d0 - z))
    else if (z <= (-2.65d-275)) then
        tmp = t_1
    else if (z <= 8.8d-183) then
        tmp = (y * x) / (y + (z * (b - y)))
    else if (z <= 4.4d-79) then
        tmp = t_1
    else
        tmp = (t / (b - y)) - (a / (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 t_1 = ((y * x) - (z * a)) / y;
	double tmp;
	if (z <= -7.5e-6) {
		tmp = (t - a) / (b - y);
	} else if (z <= -9.6e-164) {
		tmp = x * (1.0 / (1.0 - z));
	} else if (z <= -2.65e-275) {
		tmp = t_1;
	} else if (z <= 8.8e-183) {
		tmp = (y * x) / (y + (z * (b - y)));
	} else if (z <= 4.4e-79) {
		tmp = t_1;
	} else {
		tmp = (t / (b - y)) - (a / (b - y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = ((y * x) - (z * a)) / y
	tmp = 0
	if z <= -7.5e-6:
		tmp = (t - a) / (b - y)
	elif z <= -9.6e-164:
		tmp = x * (1.0 / (1.0 - z))
	elif z <= -2.65e-275:
		tmp = t_1
	elif z <= 8.8e-183:
		tmp = (y * x) / (y + (z * (b - y)))
	elif z <= 4.4e-79:
		tmp = t_1
	else:
		tmp = (t / (b - y)) - (a / (b - y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y * x) - Float64(z * a)) / y)
	tmp = 0.0
	if (z <= -7.5e-6)
		tmp = Float64(Float64(t - a) / Float64(b - y));
	elseif (z <= -9.6e-164)
		tmp = Float64(x * Float64(1.0 / Float64(1.0 - z)));
	elseif (z <= -2.65e-275)
		tmp = t_1;
	elseif (z <= 8.8e-183)
		tmp = Float64(Float64(y * x) / Float64(y + Float64(z * Float64(b - y))));
	elseif (z <= 4.4e-79)
		tmp = t_1;
	else
		tmp = Float64(Float64(t / Float64(b - y)) - Float64(a / Float64(b - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((y * x) - (z * a)) / y;
	tmp = 0.0;
	if (z <= -7.5e-6)
		tmp = (t - a) / (b - y);
	elseif (z <= -9.6e-164)
		tmp = x * (1.0 / (1.0 - z));
	elseif (z <= -2.65e-275)
		tmp = t_1;
	elseif (z <= 8.8e-183)
		tmp = (y * x) / (y + (z * (b - y)));
	elseif (z <= 4.4e-79)
		tmp = t_1;
	else
		tmp = (t / (b - y)) - (a / (b - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y * x), $MachinePrecision] - N[(z * a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[z, -7.5e-6], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -9.6e-164], N[(x * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -2.65e-275], t$95$1, If[LessEqual[z, 8.8e-183], N[(N[(y * x), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.4e-79], t$95$1, N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -7.50000000000000019e-6

   1. Initial program 41.6%

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

   2. Taylor expanded in z around inf 77.7%

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

   if -7.50000000000000019e-6 < z < -9.59999999999999932e-164

   1. Initial program 83.5%

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

   2. Taylor expanded in y around inf 52.9%

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

      1. +-commutative 52.9%

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

      2. mul-1-neg 52.9%

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

      3. unsub-neg 52.9%

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

   4. Simplified 52.9%

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

      1. div-inv 53.0%

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

   6. Applied egg-rr 53.0%

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

   if -9.59999999999999932e-164 < z < -2.64999999999999993e-275 or 8.7999999999999999e-183 < z < 4.3999999999999998e-79

   1. Initial program 92.1%

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

   2. Taylor expanded in b around inf 92.1%

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

   3. Taylor expanded in t around 0 70.6%

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

      1. *-commutative 70.6%

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

      2. mul-1-neg 70.6%

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

      3. unsub-neg 70.6%

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

      4. *-commutative 70.6%

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

   5. Simplified 70.6%

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

   6. Taylor expanded in b around 0 60.7%

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

   if -2.64999999999999993e-275 < z < 8.7999999999999999e-183

   1. Initial program 90.7%

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

      1. fma-def 90.7%

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

      2. +-commutative 90.7%

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

      3. fma-def 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in x around inf 64.7%

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

   if 4.3999999999999998e-79 < z

   1. Initial program 49.9%

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

   2. Taylor expanded in z around -inf 67.6%

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

      1. associate--l+ 67.6%

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

      2. mul-1-neg 67.6%

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

      3. distribute-lft-out-- 67.6%

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

      4. associate-/l* 70.1%

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

      5. associate-/l* 83.7%

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

   4. Simplified 83.7%

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

   5. Taylor expanded in y around inf 73.3%

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

   6. Taylor expanded in x around 0 65.1%

      \[\leadsto \color{blue}{\frac{t}{b - y} - \frac{a}{b - y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{-164}:\\ \;\;\;\;x \cdot \frac{1}{1 - z}\\ \mathbf{elif}\;z \leq -2.65 \cdot 10^{-275}:\\ \;\;\;\;\frac{y \cdot x - z \cdot a}{y}\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-183}:\\ \;\;\;\;\frac{y \cdot x}{y + z \cdot \left(b - y\right)}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-79}:\\ \;\;\;\;\frac{y \cdot x - z \cdot a}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \end{array} \]

Alternative 6: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+33} \lor \neg \left(z \leq 1.2 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{t}{b - y} - \left(\frac{x}{z} + \frac{a}{b - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -3.3e+33) (not (<= z 1.2e+19)))
   (- (/ t (- b y)) (+ (/ x z) (/ a (- b y))))
   (/ (+ (* z (- t a)) (* y x)) (+ y (* z (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -3.3e+33) || !(z <= 1.2e+19)) {
		tmp = (t / (b - y)) - ((x / z) + (a / (b - y)));
	} else {
		tmp = ((z * (t - a)) + (y * x)) / (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 <= (-3.3d+33)) .or. (.not. (z <= 1.2d+19))) then
        tmp = (t / (b - y)) - ((x / z) + (a / (b - y)))
    else
        tmp = ((z * (t - a)) + (y * x)) / (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 <= -3.3e+33) || !(z <= 1.2e+19)) {
		tmp = (t / (b - y)) - ((x / z) + (a / (b - y)));
	} else {
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * (b - y)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -3.3e+33) or not (z <= 1.2e+19):
		tmp = (t / (b - y)) - ((x / z) + (a / (b - y)))
	else:
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * (b - y)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -3.3e+33) || !(z <= 1.2e+19))
		tmp = Float64(Float64(t / Float64(b - y)) - Float64(Float64(x / z) + Float64(a / Float64(b - y))));
	else
		tmp = Float64(Float64(Float64(z * Float64(t - a)) + Float64(y * x)) / 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 <= -3.3e+33) || ~((z <= 1.2e+19)))
		tmp = (t / (b - y)) - ((x / z) + (a / (b - y)));
	else
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * (b - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -3.3e+33], N[Not[LessEqual[z, 1.2e+19]], $MachinePrecision]], N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(N[(x / z), $MachinePrecision] + N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.29999999999999976e33 or 1.2e19 < z

   1. Initial program 35.7%

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

   2. Taylor expanded in z around -inf 64.1%

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

      1. associate--l+ 64.1%

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

      2. mul-1-neg 64.1%

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

      3. distribute-lft-out-- 64.1%

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

      4. associate-/l* 72.1%

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

      5. associate-/l* 92.7%

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

   4. Simplified 92.7%

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

   5. Taylor expanded in y around inf 90.0%

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

   if -3.29999999999999976e33 < z < 1.2e19

   1. Initial program 90.0%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+33} \lor \neg \left(z \leq 1.2 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{t}{b - y} - \left(\frac{x}{z} + \frac{a}{b - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot \left(b - y\right)}\\ \end{array} \]

Alternative 7: 61.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -0.00013:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{-169}:\\ \;\;\;\;x \cdot \frac{1}{1 - z}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-122}:\\ \;\;\;\;\frac{y \cdot x - z \cdot a}{y}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-26}:\\ \;\;\;\;\frac{z \cdot t}{y + z \cdot b}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+19}:\\ \;\;\;\;\frac{z}{y} \cdot \frac{a}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -0.00013)
     t_1
     (if (<= z -4.7e-169)
       (* x (/ 1.0 (- 1.0 z)))
       (if (<= z 7e-122)
         (/ (- (* y x) (* z a)) y)
         (if (<= z 4e-26)
           (/ (* z t) (+ y (* z b)))
           (if (<= z 3.1e+19) (* (/ z y) (/ a (+ z -1.0))) t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -0.00013) {
		tmp = t_1;
	} else if (z <= -4.7e-169) {
		tmp = x * (1.0 / (1.0 - z));
	} else if (z <= 7e-122) {
		tmp = ((y * x) - (z * a)) / y;
	} else if (z <= 4e-26) {
		tmp = (z * t) / (y + (z * b));
	} else if (z <= 3.1e+19) {
		tmp = (z / y) * (a / (z + -1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-0.00013d0)) then
        tmp = t_1
    else if (z <= (-4.7d-169)) then
        tmp = x * (1.0d0 / (1.0d0 - z))
    else if (z <= 7d-122) then
        tmp = ((y * x) - (z * a)) / y
    else if (z <= 4d-26) then
        tmp = (z * t) / (y + (z * b))
    else if (z <= 3.1d+19) then
        tmp = (z / y) * (a / (z + (-1.0d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -0.00013) {
		tmp = t_1;
	} else if (z <= -4.7e-169) {
		tmp = x * (1.0 / (1.0 - z));
	} else if (z <= 7e-122) {
		tmp = ((y * x) - (z * a)) / y;
	} else if (z <= 4e-26) {
		tmp = (z * t) / (y + (z * b));
	} else if (z <= 3.1e+19) {
		tmp = (z / y) * (a / (z + -1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -0.00013:
		tmp = t_1
	elif z <= -4.7e-169:
		tmp = x * (1.0 / (1.0 - z))
	elif z <= 7e-122:
		tmp = ((y * x) - (z * a)) / y
	elif z <= 4e-26:
		tmp = (z * t) / (y + (z * b))
	elif z <= 3.1e+19:
		tmp = (z / y) * (a / (z + -1.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -0.00013)
		tmp = t_1;
	elseif (z <= -4.7e-169)
		tmp = Float64(x * Float64(1.0 / Float64(1.0 - z)));
	elseif (z <= 7e-122)
		tmp = Float64(Float64(Float64(y * x) - Float64(z * a)) / y);
	elseif (z <= 4e-26)
		tmp = Float64(Float64(z * t) / Float64(y + Float64(z * b)));
	elseif (z <= 3.1e+19)
		tmp = Float64(Float64(z / y) * Float64(a / Float64(z + -1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -0.00013)
		tmp = t_1;
	elseif (z <= -4.7e-169)
		tmp = x * (1.0 / (1.0 - z));
	elseif (z <= 7e-122)
		tmp = ((y * x) - (z * a)) / y;
	elseif (z <= 4e-26)
		tmp = (z * t) / (y + (z * b));
	elseif (z <= 3.1e+19)
		tmp = (z / y) * (a / (z + -1.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.00013], t$95$1, If[LessEqual[z, -4.7e-169], N[(x * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7e-122], N[(N[(N[(y * x), $MachinePrecision] - N[(z * a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, 4e-26], N[(N[(z * t), $MachinePrecision] / N[(y + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.1e+19], N[(N[(z / y), $MachinePrecision] * N[(a / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -1.29999999999999989e-4 or 3.1e19 < z

   1. Initial program 40.3%

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

   2. Taylor expanded in z around inf 75.2%

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

   if -1.29999999999999989e-4 < z < -4.6999999999999999e-169

   1. Initial program 83.5%

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

   2. Taylor expanded in y around inf 52.9%

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

      1. +-commutative 52.9%

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

      2. mul-1-neg 52.9%

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

      3. unsub-neg 52.9%

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

   4. Simplified 52.9%

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

      1. div-inv 53.0%

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

   6. Applied egg-rr 53.0%

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

   if -4.6999999999999999e-169 < z < 7.0000000000000003e-122

   1. Initial program 91.7%

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

   2. Taylor expanded in b around inf 91.7%

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

   3. Taylor expanded in t around 0 75.5%

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

      1. *-commutative 75.5%

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

      2. mul-1-neg 75.5%

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

      3. unsub-neg 75.5%

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

      4. *-commutative 75.5%

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

   5. Simplified 75.5%

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

   6. Taylor expanded in b around 0 60.5%

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

   if 7.0000000000000003e-122 < z < 4.0000000000000002e-26

   1. Initial program 84.9%

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

   2. Taylor expanded in b around inf 84.9%

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

   3. Taylor expanded in t around inf 52.6%

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

   if 4.0000000000000002e-26 < z < 3.1e19

   1. Initial program 88.9%

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

   2. Taylor expanded in y around -inf 77.4%

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

      1. +-commutative 77.4%

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

      2. mul-1-neg 77.4%

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

      3. unsub-neg 77.4%

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

   4. Simplified 88.4%

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

   5. Taylor expanded in b around 0 78.8%

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

      1. times-frac 89.4%

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

      2. sub-neg 89.4%

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

      3. metadata-eval 89.4%

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

   7. Simplified 89.4%

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

   8. Taylor expanded in a around inf 44.7%

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

      1. times-frac 55.7%

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

      2. sub-neg 55.7%

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

      3. metadata-eval 55.7%

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

      4. +-commutative 55.7%

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

   10. Simplified 55.7%

       \[\leadsto \color{blue}{\frac{a}{-1 + z} \cdot \frac{z}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.00013:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{-169}:\\ \;\;\;\;x \cdot \frac{1}{1 - z}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-122}:\\ \;\;\;\;\frac{y \cdot x - z \cdot a}{y}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-26}:\\ \;\;\;\;\frac{z \cdot t}{y + z \cdot b}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+19}:\\ \;\;\;\;\frac{z}{y} \cdot \frac{a}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]

Alternative 8: 71.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.00033:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-94}:\\ \;\;\;\;x + z \cdot \left(x + \frac{t - a}{y}\right)\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+15}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -0.00033)
   (/ (- t a) (- b y))
   (if (<= z 1.2e-94)
     (+ x (* z (+ x (/ (- t a) y))))
     (if (<= z 1.85e+15)
       (/ (* z (- t a)) (+ y (* z (- b y))))
       (- (/ t (- b y)) (/ a (- b y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -0.00033) {
		tmp = (t - a) / (b - y);
	} else if (z <= 1.2e-94) {
		tmp = x + (z * (x + ((t - a) / y)));
	} else if (z <= 1.85e+15) {
		tmp = (z * (t - a)) / (y + (z * (b - y)));
	} else {
		tmp = (t / (b - y)) - (a / (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 <= (-0.00033d0)) then
        tmp = (t - a) / (b - y)
    else if (z <= 1.2d-94) then
        tmp = x + (z * (x + ((t - a) / y)))
    else if (z <= 1.85d+15) then
        tmp = (z * (t - a)) / (y + (z * (b - y)))
    else
        tmp = (t / (b - y)) - (a / (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 <= -0.00033) {
		tmp = (t - a) / (b - y);
	} else if (z <= 1.2e-94) {
		tmp = x + (z * (x + ((t - a) / y)));
	} else if (z <= 1.85e+15) {
		tmp = (z * (t - a)) / (y + (z * (b - y)));
	} else {
		tmp = (t / (b - y)) - (a / (b - y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -0.00033:
		tmp = (t - a) / (b - y)
	elif z <= 1.2e-94:
		tmp = x + (z * (x + ((t - a) / y)))
	elif z <= 1.85e+15:
		tmp = (z * (t - a)) / (y + (z * (b - y)))
	else:
		tmp = (t / (b - y)) - (a / (b - y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -0.00033)
		tmp = Float64(Float64(t - a) / Float64(b - y));
	elseif (z <= 1.2e-94)
		tmp = Float64(x + Float64(z * Float64(x + Float64(Float64(t - a) / y))));
	elseif (z <= 1.85e+15)
		tmp = Float64(Float64(z * Float64(t - a)) / Float64(y + Float64(z * Float64(b - y))));
	else
		tmp = Float64(Float64(t / Float64(b - y)) - Float64(a / Float64(b - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -0.00033)
		tmp = (t - a) / (b - y);
	elseif (z <= 1.2e-94)
		tmp = x + (z * (x + ((t - a) / y)));
	elseif (z <= 1.85e+15)
		tmp = (z * (t - a)) / (y + (z * (b - y)));
	else
		tmp = (t / (b - y)) - (a / (b - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -0.00033], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.2e-94], N[(x + N[(z * N[(x + N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.85e+15], N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.3e-4

   1. Initial program 41.6%

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

   2. Taylor expanded in z around inf 77.7%

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

   if -3.3e-4 < z < 1.2e-94

   1. Initial program 89.1%

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

   2. Taylor expanded in y around -inf 56.9%

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

      1. +-commutative 56.9%

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

      2. mul-1-neg 56.9%

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

      3. unsub-neg 56.9%

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

   4. Simplified 70.9%

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

   5. Taylor expanded in b around 0 78.9%

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

      1. times-frac 74.2%

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

      2. sub-neg 74.2%

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

      3. metadata-eval 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in z around 0 73.4%

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

   if 1.2e-94 < z < 1.85e15

   1. Initial program 90.4%

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

      1. fma-def 90.4%

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

      2. +-commutative 90.4%

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

      3. fma-def 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in x around 0 71.1%

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

   if 1.85e15 < z

   1. Initial program 38.1%

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

   2. Taylor expanded in z around -inf 70.3%

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

      1. associate--l+ 70.3%

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

      2. mul-1-neg 70.3%

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

      3. distribute-lft-out-- 70.3%

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

      4. associate-/l* 73.6%

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

      5. associate-/l* 91.3%

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

   4. Simplified 91.3%

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

   5. Taylor expanded in y around inf 87.9%

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

   6. Taylor expanded in x around 0 72.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.00033:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-94}:\\ \;\;\;\;x + z \cdot \left(x + \frac{t - a}{y}\right)\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{+15}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \end{array} \]

Alternative 9: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+30} \lor \neg \left(z \leq 50\right):\\ \;\;\;\;\frac{t}{b - y} - \left(\frac{x}{z} + \frac{a}{b - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.5e+30) (not (<= z 50.0)))
   (- (/ t (- b y)) (+ (/ x z) (/ a (- b y))))
   (/ (+ (* z (- t a)) (* y x)) (+ y (* z b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.5e+30) || !(z <= 50.0)) {
		tmp = (t / (b - y)) - ((x / z) + (a / (b - y)));
	} else {
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * b));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((z <= (-1.5d+30)) .or. (.not. (z <= 50.0d0))) then
        tmp = (t / (b - y)) - ((x / z) + (a / (b - y)))
    else
        tmp = ((z * (t - a)) + (y * x)) / (y + (z * b))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.5e+30) || !(z <= 50.0)) {
		tmp = (t / (b - y)) - ((x / z) + (a / (b - y)));
	} else {
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * b));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -1.5e+30) or not (z <= 50.0):
		tmp = (t / (b - y)) - ((x / z) + (a / (b - y)))
	else:
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * b))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.5e+30) || !(z <= 50.0))
		tmp = Float64(Float64(t / Float64(b - y)) - Float64(Float64(x / z) + Float64(a / Float64(b - y))));
	else
		tmp = Float64(Float64(Float64(z * Float64(t - a)) + Float64(y * x)) / Float64(y + Float64(z * b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -1.5e+30) || ~((z <= 50.0)))
		tmp = (t / (b - y)) - ((x / z) + (a / (b - y)));
	else
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * b));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.5e+30], N[Not[LessEqual[z, 50.0]], $MachinePrecision]], N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(N[(x / z), $MachinePrecision] + N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.49999999999999989e30 or 50 < z

   1. Initial program 37.7%

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

   2. Taylor expanded in z around -inf 64.4%

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

      1. associate--l+ 64.4%

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

      2. mul-1-neg 64.4%

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

      3. distribute-lft-out-- 64.4%

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

      4. associate-/l* 72.0%

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

      5. associate-/l* 92.0%

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

   4. Simplified 92.0%

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

   5. Taylor expanded in y around inf 88.7%

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

   if -1.49999999999999989e30 < z < 50

   1. Initial program 89.7%

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

   2. Taylor expanded in b around inf 87.6%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+30} \lor \neg \left(z \leq 50\right):\\ \;\;\;\;\frac{t}{b - y} - \left(\frac{x}{z} + \frac{a}{b - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot b}\\ \end{array} \]

Alternative 10: 82.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 7.3:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.3e+17)
   (/ (- t a) (- b y))
   (if (<= z 7.3)
     (/ (+ (* z (- t a)) (* y x)) (+ y (* z b)))
     (- (/ t (- b y)) (/ a (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.3e+17) {
		tmp = (t - a) / (b - y);
	} else if (z <= 7.3) {
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * b));
	} else {
		tmp = (t / (b - y)) - (a / (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 <= (-1.3d+17)) then
        tmp = (t - a) / (b - y)
    else if (z <= 7.3d0) then
        tmp = ((z * (t - a)) + (y * x)) / (y + (z * b))
    else
        tmp = (t / (b - y)) - (a / (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 <= -1.3e+17) {
		tmp = (t - a) / (b - y);
	} else if (z <= 7.3) {
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * b));
	} else {
		tmp = (t / (b - y)) - (a / (b - y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -1.3e+17:
		tmp = (t - a) / (b - y)
	elif z <= 7.3:
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * b))
	else:
		tmp = (t / (b - y)) - (a / (b - y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.3e+17)
		tmp = Float64(Float64(t - a) / Float64(b - y));
	elseif (z <= 7.3)
		tmp = Float64(Float64(Float64(z * Float64(t - a)) + Float64(y * x)) / Float64(y + Float64(z * b)));
	else
		tmp = Float64(Float64(t / Float64(b - y)) - Float64(a / Float64(b - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -1.3e+17)
		tmp = (t - a) / (b - y);
	elseif (z <= 7.3)
		tmp = ((z * (t - a)) + (y * x)) / (y + (z * b));
	else
		tmp = (t / (b - y)) - (a / (b - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.3e+17], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.3], N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3e17

   1. Initial program 36.5%

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

   2. Taylor expanded in z around inf 77.1%

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

   if -1.3e17 < z < 7.29999999999999982

   1. Initial program 89.4%

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

   2. Taylor expanded in b around inf 88.1%

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

   if 7.29999999999999982 < z

   1. Initial program 42.2%

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

   2. Taylor expanded in z around -inf 70.4%

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

      1. associate--l+ 70.4%

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

      2. mul-1-neg 70.4%

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

      3. distribute-lft-out-- 70.4%

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

      4. associate-/l* 73.5%

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

      5. associate-/l* 90.0%

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

   4. Simplified 90.0%

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

   5. Taylor expanded in y around inf 85.3%

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

   6. Taylor expanded in x around 0 70.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 7.3:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y + z \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \end{array} \]

Alternative 11: 39.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ t_2 := \frac{t}{b - y}\\ \mathbf{if}\;t \leq -2.55 \cdot 10^{+151}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -9 \cdot 10^{+49}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.4 \cdot 10^{-296}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.62 \cdot 10^{-124}:\\ \;\;\;\;-\frac{a}{b}\\ \mathbf{elif}\;t \leq 210000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))) (t_2 (/ t (- b y))))
   (if (<= t -2.55e+151)
     t_2
     (if (<= t -5e+74)
       t_1
       (if (<= t -9e+49)
         t_2
         (if (<= t -3.4e-296)
           t_1
           (if (<= t 1.62e-124)
             (- (/ a b))
             (if (<= t 210000000000.0) t_1 t_2))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double t_2 = t / (b - y);
	double tmp;
	if (t <= -2.55e+151) {
		tmp = t_2;
	} else if (t <= -5e+74) {
		tmp = t_1;
	} else if (t <= -9e+49) {
		tmp = t_2;
	} else if (t <= -3.4e-296) {
		tmp = t_1;
	} else if (t <= 1.62e-124) {
		tmp = -(a / b);
	} else if (t <= 210000000000.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = x / (1.0d0 - z)
    t_2 = t / (b - y)
    if (t <= (-2.55d+151)) then
        tmp = t_2
    else if (t <= (-5d+74)) then
        tmp = t_1
    else if (t <= (-9d+49)) then
        tmp = t_2
    else if (t <= (-3.4d-296)) then
        tmp = t_1
    else if (t <= 1.62d-124) then
        tmp = -(a / b)
    else if (t <= 210000000000.0d0) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double t_2 = t / (b - y);
	double tmp;
	if (t <= -2.55e+151) {
		tmp = t_2;
	} else if (t <= -5e+74) {
		tmp = t_1;
	} else if (t <= -9e+49) {
		tmp = t_2;
	} else if (t <= -3.4e-296) {
		tmp = t_1;
	} else if (t <= 1.62e-124) {
		tmp = -(a / b);
	} else if (t <= 210000000000.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	t_2 = t / (b - y)
	tmp = 0
	if t <= -2.55e+151:
		tmp = t_2
	elif t <= -5e+74:
		tmp = t_1
	elif t <= -9e+49:
		tmp = t_2
	elif t <= -3.4e-296:
		tmp = t_1
	elif t <= 1.62e-124:
		tmp = -(a / b)
	elif t <= 210000000000.0:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	t_2 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (t <= -2.55e+151)
		tmp = t_2;
	elseif (t <= -5e+74)
		tmp = t_1;
	elseif (t <= -9e+49)
		tmp = t_2;
	elseif (t <= -3.4e-296)
		tmp = t_1;
	elseif (t <= 1.62e-124)
		tmp = Float64(-Float64(a / b));
	elseif (t <= 210000000000.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	t_2 = t / (b - y);
	tmp = 0.0;
	if (t <= -2.55e+151)
		tmp = t_2;
	elseif (t <= -5e+74)
		tmp = t_1;
	elseif (t <= -9e+49)
		tmp = t_2;
	elseif (t <= -3.4e-296)
		tmp = t_1;
	elseif (t <= 1.62e-124)
		tmp = -(a / b);
	elseif (t <= 210000000000.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.55e+151], t$95$2, If[LessEqual[t, -5e+74], t$95$1, If[LessEqual[t, -9e+49], t$95$2, If[LessEqual[t, -3.4e-296], t$95$1, If[LessEqual[t, 1.62e-124], (-N[(a / b), $MachinePrecision]), If[LessEqual[t, 210000000000.0], t$95$1, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.54999999999999998e151 or -4.99999999999999963e74 < t < -8.99999999999999965e49 or 2.1e11 < t

   1. Initial program 55.5%

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

   2. Taylor expanded in z around inf 65.2%

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

   3. Taylor expanded in t around inf 54.0%

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

   if -2.54999999999999998e151 < t < -4.99999999999999963e74 or -8.99999999999999965e49 < t < -3.39999999999999997e-296 or 1.62000000000000006e-124 < t < 2.1e11

   1. Initial program 69.4%

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

   2. Taylor expanded in y around inf 48.7%

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

      1. +-commutative 48.7%

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

      2. mul-1-neg 48.7%

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

      3. unsub-neg 48.7%

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

   4. Simplified 48.7%

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

   if -3.39999999999999997e-296 < t < 1.62000000000000006e-124

   1. Initial program 69.9%

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

   2. Taylor expanded in b around inf 67.3%

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

   3. Taylor expanded in t around 0 64.4%

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

      1. *-commutative 64.4%

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

      2. mul-1-neg 64.4%

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

      3. unsub-neg 64.4%

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

      4. *-commutative 64.4%

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

   5. Simplified 64.4%

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

   6. Taylor expanded in y around 0 41.0%

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

      1. associate-*r/ 41.0%

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

      2. neg-mul-1 41.0%

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

   8. Simplified 41.0%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.55 \cdot 10^{+151}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{+74}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;t \leq -9 \cdot 10^{+49}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{elif}\;t \leq -3.4 \cdot 10^{-296}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;t \leq 1.62 \cdot 10^{-124}:\\ \;\;\;\;-\frac{a}{b}\\ \mathbf{elif}\;t \leq 210000000000:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y}\\ \end{array} \]

Alternative 12: 53.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y}\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -9.8 \cdot 10^{-67}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2900000000:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{1 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) y)))
   (if (<= y -8.5e+17)
     (/ x (- 1.0 z))
     (if (<= y -1.2e-36)
       t_1
       (if (<= y -9.8e-67)
         (* x (+ z 1.0))
         (if (<= y -9e-68)
           t_1
           (if (<= y 2900000000.0) (/ (- t a) b) (* x (/ 1.0 (- 1.0 z))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / y;
	double tmp;
	if (y <= -8.5e+17) {
		tmp = x / (1.0 - z);
	} else if (y <= -1.2e-36) {
		tmp = t_1;
	} else if (y <= -9.8e-67) {
		tmp = x * (z + 1.0);
	} else if (y <= -9e-68) {
		tmp = t_1;
	} else if (y <= 2900000000.0) {
		tmp = (t - a) / b;
	} else {
		tmp = x * (1.0 / (1.0 - z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a - t) / y
    if (y <= (-8.5d+17)) then
        tmp = x / (1.0d0 - z)
    else if (y <= (-1.2d-36)) then
        tmp = t_1
    else if (y <= (-9.8d-67)) then
        tmp = x * (z + 1.0d0)
    else if (y <= (-9d-68)) then
        tmp = t_1
    else if (y <= 2900000000.0d0) then
        tmp = (t - a) / b
    else
        tmp = x * (1.0d0 / (1.0d0 - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / y;
	double tmp;
	if (y <= -8.5e+17) {
		tmp = x / (1.0 - z);
	} else if (y <= -1.2e-36) {
		tmp = t_1;
	} else if (y <= -9.8e-67) {
		tmp = x * (z + 1.0);
	} else if (y <= -9e-68) {
		tmp = t_1;
	} else if (y <= 2900000000.0) {
		tmp = (t - a) / b;
	} else {
		tmp = x * (1.0 / (1.0 - z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - t) / y
	tmp = 0
	if y <= -8.5e+17:
		tmp = x / (1.0 - z)
	elif y <= -1.2e-36:
		tmp = t_1
	elif y <= -9.8e-67:
		tmp = x * (z + 1.0)
	elif y <= -9e-68:
		tmp = t_1
	elif y <= 2900000000.0:
		tmp = (t - a) / b
	else:
		tmp = x * (1.0 / (1.0 - z))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / y)
	tmp = 0.0
	if (y <= -8.5e+17)
		tmp = Float64(x / Float64(1.0 - z));
	elseif (y <= -1.2e-36)
		tmp = t_1;
	elseif (y <= -9.8e-67)
		tmp = Float64(x * Float64(z + 1.0));
	elseif (y <= -9e-68)
		tmp = t_1;
	elseif (y <= 2900000000.0)
		tmp = Float64(Float64(t - a) / b);
	else
		tmp = Float64(x * Float64(1.0 / Float64(1.0 - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - t) / y;
	tmp = 0.0;
	if (y <= -8.5e+17)
		tmp = x / (1.0 - z);
	elseif (y <= -1.2e-36)
		tmp = t_1;
	elseif (y <= -9.8e-67)
		tmp = x * (z + 1.0);
	elseif (y <= -9e-68)
		tmp = t_1;
	elseif (y <= 2900000000.0)
		tmp = (t - a) / b;
	else
		tmp = x * (1.0 / (1.0 - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[y, -8.5e+17], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.2e-36], t$95$1, If[LessEqual[y, -9.8e-67], N[(x * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -9e-68], t$95$1, If[LessEqual[y, 2900000000.0], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], N[(x * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -8.5e17

   1. Initial program 45.6%

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

   2. Taylor expanded in y around inf 53.2%

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

      1. +-commutative 53.2%

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

      2. mul-1-neg 53.2%

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

      3. unsub-neg 53.2%

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

   4. Simplified 53.2%

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

   if -8.5e17 < y < -1.2e-36 or -9.79999999999999987e-67 < y < -8.99999999999999998e-68

   1. Initial program 50.2%

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

   2. Taylor expanded in z around inf 75.1%

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

   3. Taylor expanded in b around 0 62.6%

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

      1. mul-1-neg 62.6%

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

   5. Simplified 62.6%

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

   if -1.2e-36 < y < -9.79999999999999987e-67

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 58.5%

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

      1. +-commutative 58.5%

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

      2. mul-1-neg 58.5%

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

      3. unsub-neg 58.5%

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

   4. Simplified 58.5%

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

      1. div-inv 58.5%

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

   6. Applied egg-rr 58.5%

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

   7. Taylor expanded in z around 0 59.6%

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

      1. +-commutative 59.6%

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

   9. Simplified 59.6%

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

   if -8.99999999999999998e-68 < y < 2.9e9

   1. Initial program 81.4%

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

   2. Taylor expanded in y around 0 54.4%

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

   if 2.9e9 < y

   1. Initial program 52.9%

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

   2. Taylor expanded in y around inf 52.9%

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

      1. +-commutative 52.9%

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

      2. mul-1-neg 52.9%

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

      3. unsub-neg 52.9%

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

   4. Simplified 52.9%

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

      1. div-inv 52.9%

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

   6. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{x \cdot \frac{1}{1 - z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{a - t}{y}\\ \mathbf{elif}\;y \leq -9.8 \cdot 10^{-67}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-68}:\\ \;\;\;\;\frac{a - t}{y}\\ \mathbf{elif}\;y \leq 2900000000:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{1 - z}\\ \end{array} \]

Alternative 13: 63.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.0004:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-163}:\\ \;\;\;\;x \cdot \frac{1}{1 - z}\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-77}:\\ \;\;\;\;\frac{y \cdot x - z \cdot a}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -0.0004)
   (/ (- t a) (- b y))
   (if (<= z -1.1e-163)
     (* x (/ 1.0 (- 1.0 z)))
     (if (<= z 1.12e-77)
       (/ (- (* y x) (* z a)) y)
       (- (/ t (- b y)) (/ a (- b y)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -0.0004) {
		tmp = (t - a) / (b - y);
	} else if (z <= -1.1e-163) {
		tmp = x * (1.0 / (1.0 - z));
	} else if (z <= 1.12e-77) {
		tmp = ((y * x) - (z * a)) / y;
	} else {
		tmp = (t / (b - y)) - (a / (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 <= (-0.0004d0)) then
        tmp = (t - a) / (b - y)
    else if (z <= (-1.1d-163)) then
        tmp = x * (1.0d0 / (1.0d0 - z))
    else if (z <= 1.12d-77) then
        tmp = ((y * x) - (z * a)) / y
    else
        tmp = (t / (b - y)) - (a / (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 <= -0.0004) {
		tmp = (t - a) / (b - y);
	} else if (z <= -1.1e-163) {
		tmp = x * (1.0 / (1.0 - z));
	} else if (z <= 1.12e-77) {
		tmp = ((y * x) - (z * a)) / y;
	} else {
		tmp = (t / (b - y)) - (a / (b - y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -0.0004:
		tmp = (t - a) / (b - y)
	elif z <= -1.1e-163:
		tmp = x * (1.0 / (1.0 - z))
	elif z <= 1.12e-77:
		tmp = ((y * x) - (z * a)) / y
	else:
		tmp = (t / (b - y)) - (a / (b - y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -0.0004)
		tmp = Float64(Float64(t - a) / Float64(b - y));
	elseif (z <= -1.1e-163)
		tmp = Float64(x * Float64(1.0 / Float64(1.0 - z)));
	elseif (z <= 1.12e-77)
		tmp = Float64(Float64(Float64(y * x) - Float64(z * a)) / y);
	else
		tmp = Float64(Float64(t / Float64(b - y)) - Float64(a / Float64(b - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -0.0004)
		tmp = (t - a) / (b - y);
	elseif (z <= -1.1e-163)
		tmp = x * (1.0 / (1.0 - z));
	elseif (z <= 1.12e-77)
		tmp = ((y * x) - (z * a)) / y;
	else
		tmp = (t / (b - y)) - (a / (b - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -0.0004], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.1e-163], N[(x * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.12e-77], N[(N[(N[(y * x), $MachinePrecision] - N[(z * a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.00000000000000019e-4

   1. Initial program 41.6%

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

   2. Taylor expanded in z around inf 77.7%

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

   if -4.00000000000000019e-4 < z < -1.10000000000000005e-163

   1. Initial program 83.5%

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

   2. Taylor expanded in y around inf 52.9%

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

      1. +-commutative 52.9%

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

      2. mul-1-neg 52.9%

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

      3. unsub-neg 52.9%

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

   4. Simplified 52.9%

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

      1. div-inv 53.0%

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

   6. Applied egg-rr 53.0%

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

   if -1.10000000000000005e-163 < z < 1.12000000000000009e-77

   1. Initial program 91.5%

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

   2. Taylor expanded in b around inf 91.5%

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

   3. Taylor expanded in t around 0 71.3%

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

      1. *-commutative 71.3%

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

      2. mul-1-neg 71.3%

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

      3. unsub-neg 71.3%

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

      4. *-commutative 71.3%

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

   5. Simplified 71.3%

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

   6. Taylor expanded in b around 0 55.8%

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

   if 1.12000000000000009e-77 < z

   1. Initial program 49.9%

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

   2. Taylor expanded in z around -inf 67.6%

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

      1. associate--l+ 67.6%

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

      2. mul-1-neg 67.6%

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

      3. distribute-lft-out-- 67.6%

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

      4. associate-/l* 70.1%

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

      5. associate-/l* 83.7%

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

   4. Simplified 83.7%

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

   5. Taylor expanded in y around inf 73.3%

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

   6. Taylor expanded in x around 0 65.1%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.0004:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-163}:\\ \;\;\;\;x \cdot \frac{1}{1 - z}\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{-77}:\\ \;\;\;\;\frac{y \cdot x - z \cdot a}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \end{array} \]

Alternative 14: 53.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y}\\ t_2 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{+18}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-66}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1900000000:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) y)) (t_2 (/ x (- 1.0 z))))
   (if (<= y -1.45e+18)
     t_2
     (if (<= y -3.4e-36)
       t_1
       (if (<= y -1.15e-66)
         (* x (+ z 1.0))
         (if (<= y -5.2e-68)
           t_1
           (if (<= y 1900000000.0) (/ (- t a) b) t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / y;
	double t_2 = x / (1.0 - z);
	double tmp;
	if (y <= -1.45e+18) {
		tmp = t_2;
	} else if (y <= -3.4e-36) {
		tmp = t_1;
	} else if (y <= -1.15e-66) {
		tmp = x * (z + 1.0);
	} else if (y <= -5.2e-68) {
		tmp = t_1;
	} else if (y <= 1900000000.0) {
		tmp = (t - a) / b;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (a - t) / y
    t_2 = x / (1.0d0 - z)
    if (y <= (-1.45d+18)) then
        tmp = t_2
    else if (y <= (-3.4d-36)) then
        tmp = t_1
    else if (y <= (-1.15d-66)) then
        tmp = x * (z + 1.0d0)
    else if (y <= (-5.2d-68)) then
        tmp = t_1
    else if (y <= 1900000000.0d0) then
        tmp = (t - a) / b
    else
        tmp = t_2
    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 - t) / y;
	double t_2 = x / (1.0 - z);
	double tmp;
	if (y <= -1.45e+18) {
		tmp = t_2;
	} else if (y <= -3.4e-36) {
		tmp = t_1;
	} else if (y <= -1.15e-66) {
		tmp = x * (z + 1.0);
	} else if (y <= -5.2e-68) {
		tmp = t_1;
	} else if (y <= 1900000000.0) {
		tmp = (t - a) / b;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - t) / y
	t_2 = x / (1.0 - z)
	tmp = 0
	if y <= -1.45e+18:
		tmp = t_2
	elif y <= -3.4e-36:
		tmp = t_1
	elif y <= -1.15e-66:
		tmp = x * (z + 1.0)
	elif y <= -5.2e-68:
		tmp = t_1
	elif y <= 1900000000.0:
		tmp = (t - a) / b
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / y)
	t_2 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -1.45e+18)
		tmp = t_2;
	elseif (y <= -3.4e-36)
		tmp = t_1;
	elseif (y <= -1.15e-66)
		tmp = Float64(x * Float64(z + 1.0));
	elseif (y <= -5.2e-68)
		tmp = t_1;
	elseif (y <= 1900000000.0)
		tmp = Float64(Float64(t - a) / b);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - t) / y;
	t_2 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -1.45e+18)
		tmp = t_2;
	elseif (y <= -3.4e-36)
		tmp = t_1;
	elseif (y <= -1.15e-66)
		tmp = x * (z + 1.0);
	elseif (y <= -5.2e-68)
		tmp = t_1;
	elseif (y <= 1900000000.0)
		tmp = (t - a) / b;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.45e+18], t$95$2, If[LessEqual[y, -3.4e-36], t$95$1, If[LessEqual[y, -1.15e-66], N[(x * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.2e-68], t$95$1, If[LessEqual[y, 1900000000.0], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.45e18 or 1.9e9 < y

   1. Initial program 48.8%

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

   2. Taylor expanded in y around inf 53.1%

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

      1. +-commutative 53.1%

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

      2. mul-1-neg 53.1%

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

      3. unsub-neg 53.1%

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

   4. Simplified 53.1%

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

   if -1.45e18 < y < -3.4000000000000003e-36 or -1.14999999999999996e-66 < y < -5.1999999999999996e-68

   1. Initial program 50.2%

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

   2. Taylor expanded in z around inf 75.1%

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

   3. Taylor expanded in b around 0 62.6%

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

      1. mul-1-neg 62.6%

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

   5. Simplified 62.6%

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

   if -3.4000000000000003e-36 < y < -1.14999999999999996e-66

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 58.5%

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

      1. +-commutative 58.5%

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

      2. mul-1-neg 58.5%

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

      3. unsub-neg 58.5%

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

   4. Simplified 58.5%

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

      1. div-inv 58.5%

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

   6. Applied egg-rr 58.5%

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

   7. Taylor expanded in z around 0 59.6%

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

      1. +-commutative 59.6%

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

   9. Simplified 59.6%

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

   if -5.1999999999999996e-68 < y < 1.9e9

   1. Initial program 81.4%

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

   2. Taylor expanded in y around 0 54.4%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{+18}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;y \leq -3.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{a - t}{y}\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-66}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-68}:\\ \;\;\;\;\frac{a - t}{y}\\ \mathbf{elif}\;y \leq 1900000000:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - z}\\ \end{array} \]

Alternative 15: 62.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -0.017:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-118}:\\ \;\;\;\;x \cdot \frac{1}{1 - z}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-29}:\\ \;\;\;\;\frac{z \cdot t}{y + z \cdot b}\\ \mathbf{elif}\;z \leq 460:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -0.017)
     t_1
     (if (<= z 9.8e-118)
       (* x (/ 1.0 (- 1.0 z)))
       (if (<= z 8e-29)
         (/ (* z t) (+ y (* z b)))
         (if (<= z 460.0) (/ 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 - a) / (b - y);
	double tmp;
	if (z <= -0.017) {
		tmp = t_1;
	} else if (z <= 9.8e-118) {
		tmp = x * (1.0 / (1.0 - z));
	} else if (z <= 8e-29) {
		tmp = (z * t) / (y + (z * b));
	} else if (z <= 460.0) {
		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 - a) / (b - y)
    if (z <= (-0.017d0)) then
        tmp = t_1
    else if (z <= 9.8d-118) then
        tmp = x * (1.0d0 / (1.0d0 - z))
    else if (z <= 8d-29) then
        tmp = (z * t) / (y + (z * b))
    else if (z <= 460.0d0) 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 - a) / (b - y);
	double tmp;
	if (z <= -0.017) {
		tmp = t_1;
	} else if (z <= 9.8e-118) {
		tmp = x * (1.0 / (1.0 - z));
	} else if (z <= 8e-29) {
		tmp = (z * t) / (y + (z * b));
	} else if (z <= 460.0) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -0.017:
		tmp = t_1
	elif z <= 9.8e-118:
		tmp = x * (1.0 / (1.0 - z))
	elif z <= 8e-29:
		tmp = (z * t) / (y + (z * b))
	elif z <= 460.0:
		tmp = x / (1.0 - z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -0.017)
		tmp = t_1;
	elseif (z <= 9.8e-118)
		tmp = Float64(x * Float64(1.0 / Float64(1.0 - z)));
	elseif (z <= 8e-29)
		tmp = Float64(Float64(z * t) / Float64(y + Float64(z * b)));
	elseif (z <= 460.0)
		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 - a) / (b - y);
	tmp = 0.0;
	if (z <= -0.017)
		tmp = t_1;
	elseif (z <= 9.8e-118)
		tmp = x * (1.0 / (1.0 - z));
	elseif (z <= 8e-29)
		tmp = (z * t) / (y + (z * b));
	elseif (z <= 460.0)
		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[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.017], t$95$1, If[LessEqual[z, 9.8e-118], N[(x * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8e-29], N[(N[(z * t), $MachinePrecision] / N[(y + N[(z * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 460.0], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -0.017000000000000001 or 460 < z

   1. Initial program 41.4%

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

   2. Taylor expanded in z around inf 75.0%

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

   if -0.017000000000000001 < z < 9.7999999999999995e-118

   1. Initial program 88.6%

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

   2. Taylor expanded in y around inf 50.1%

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

      1. +-commutative 50.1%

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

      2. mul-1-neg 50.1%

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

      3. unsub-neg 50.1%

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

   4. Simplified 50.1%

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

      1. div-inv 50.1%

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

   6. Applied egg-rr 50.1%

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

   if 9.7999999999999995e-118 < z < 7.99999999999999955e-29

   1. Initial program 93.8%

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

   2. Taylor expanded in b around inf 93.8%

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

   3. Taylor expanded in t around inf 56.1%

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

   if 7.99999999999999955e-29 < z < 460

   1. Initial program 84.0%

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

   2. Taylor expanded in y around inf 52.8%

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

      1. +-commutative 52.8%

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

      2. mul-1-neg 52.8%

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

      3. unsub-neg 52.8%

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

   4. Simplified 52.8%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.017:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{-118}:\\ \;\;\;\;x \cdot \frac{1}{1 - z}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-29}:\\ \;\;\;\;\frac{z \cdot t}{y + z \cdot b}\\ \mathbf{elif}\;z \leq 460:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]

Alternative 16: 68.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-77}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -5.7e-6)
   (/ (- t a) (- b y))
   (if (<= z 1.15e-77)
     (/ (+ (* z (- t a)) (* y x)) y)
     (- (/ t (- b y)) (/ a (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -5.7e-6) {
		tmp = (t - a) / (b - y);
	} else if (z <= 1.15e-77) {
		tmp = ((z * (t - a)) + (y * x)) / y;
	} else {
		tmp = (t / (b - y)) - (a / (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 <= (-5.7d-6)) then
        tmp = (t - a) / (b - y)
    else if (z <= 1.15d-77) then
        tmp = ((z * (t - a)) + (y * x)) / y
    else
        tmp = (t / (b - y)) - (a / (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 <= -5.7e-6) {
		tmp = (t - a) / (b - y);
	} else if (z <= 1.15e-77) {
		tmp = ((z * (t - a)) + (y * x)) / y;
	} else {
		tmp = (t / (b - y)) - (a / (b - y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -5.7e-6:
		tmp = (t - a) / (b - y)
	elif z <= 1.15e-77:
		tmp = ((z * (t - a)) + (y * x)) / y
	else:
		tmp = (t / (b - y)) - (a / (b - y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -5.7e-6)
		tmp = Float64(Float64(t - a) / Float64(b - y));
	elseif (z <= 1.15e-77)
		tmp = Float64(Float64(Float64(z * Float64(t - a)) + Float64(y * x)) / y);
	else
		tmp = Float64(Float64(t / Float64(b - y)) - Float64(a / Float64(b - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -5.7e-6)
		tmp = (t - a) / (b - y);
	elseif (z <= 1.15e-77)
		tmp = ((z * (t - a)) + (y * x)) / y;
	else
		tmp = (t / (b - y)) - (a / (b - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -5.7e-6], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.15e-77], N[(N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.6999999999999996e-6

   1. Initial program 41.6%

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

   2. Taylor expanded in z around inf 77.7%

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

   if -5.6999999999999996e-6 < z < 1.14999999999999999e-77

   1. Initial program 89.4%

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

   2. Taylor expanded in b around inf 88.3%

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

   3. Taylor expanded in b around 0 68.9%

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

   if 1.14999999999999999e-77 < z

   1. Initial program 49.9%

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

   2. Taylor expanded in z around -inf 67.6%

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

      1. associate--l+ 67.6%

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

      2. mul-1-neg 67.6%

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

      3. distribute-lft-out-- 67.6%

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

      4. associate-/l* 70.1%

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

      5. associate-/l* 83.7%

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

   4. Simplified 83.7%

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

   5. Taylor expanded in y around inf 73.3%

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

   6. Taylor expanded in x around 0 65.1%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-77}:\\ \;\;\;\;\frac{z \cdot \left(t - a\right) + y \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \end{array} \]

Alternative 17: 71.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.00037:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 0.00012:\\ \;\;\;\;x + z \cdot \left(x + \frac{t - a}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -0.00037)
   (/ (- t a) (- b y))
   (if (<= z 0.00012)
     (+ x (* z (+ x (/ (- t a) y))))
     (- (/ t (- b y)) (/ a (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -0.00037) {
		tmp = (t - a) / (b - y);
	} else if (z <= 0.00012) {
		tmp = x + (z * (x + ((t - a) / y)));
	} else {
		tmp = (t / (b - y)) - (a / (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 <= (-0.00037d0)) then
        tmp = (t - a) / (b - y)
    else if (z <= 0.00012d0) then
        tmp = x + (z * (x + ((t - a) / y)))
    else
        tmp = (t / (b - y)) - (a / (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 <= -0.00037) {
		tmp = (t - a) / (b - y);
	} else if (z <= 0.00012) {
		tmp = x + (z * (x + ((t - a) / y)));
	} else {
		tmp = (t / (b - y)) - (a / (b - y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -0.00037:
		tmp = (t - a) / (b - y)
	elif z <= 0.00012:
		tmp = x + (z * (x + ((t - a) / y)))
	else:
		tmp = (t / (b - y)) - (a / (b - y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -0.00037)
		tmp = Float64(Float64(t - a) / Float64(b - y));
	elseif (z <= 0.00012)
		tmp = Float64(x + Float64(z * Float64(x + Float64(Float64(t - a) / y))));
	else
		tmp = Float64(Float64(t / Float64(b - y)) - Float64(a / Float64(b - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -0.00037)
		tmp = (t - a) / (b - y);
	elseif (z <= 0.00012)
		tmp = x + (z * (x + ((t - a) / y)));
	else
		tmp = (t / (b - y)) - (a / (b - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -0.00037], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.00012], N[(x + N[(z * N[(x + N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.6999999999999999e-4

   1. Initial program 41.6%

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

   2. Taylor expanded in z around inf 77.7%

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

   if -3.6999999999999999e-4 < z < 1.20000000000000003e-4

   1. Initial program 89.0%

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

   2. Taylor expanded in y around -inf 56.0%

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

      1. +-commutative 56.0%

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

      2. mul-1-neg 56.0%

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

      3. unsub-neg 56.0%

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

   4. Simplified 69.0%

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

   5. Taylor expanded in b around 0 76.1%

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

      1. times-frac 71.9%

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

      2. sub-neg 71.9%

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

      3. metadata-eval 71.9%

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

   7. Simplified 71.9%

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

   8. Taylor expanded in z around 0 70.2%

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

   if 1.20000000000000003e-4 < z

   1. Initial program 42.2%

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

   2. Taylor expanded in z around -inf 70.4%

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

      1. associate--l+ 70.4%

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

      2. mul-1-neg 70.4%

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

      3. distribute-lft-out-- 70.4%

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

      4. associate-/l* 73.5%

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

      5. associate-/l* 90.0%

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

   4. Simplified 90.0%

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

   5. Taylor expanded in y around inf 85.3%

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

   6. Taylor expanded in x around 0 70.4%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.00037:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq 0.00012:\\ \;\;\;\;x + z \cdot \left(x + \frac{t - a}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \end{array} \]

Alternative 18: 43.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -7 \cdot 10^{+125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+54}:\\ \;\;\;\;-\frac{x}{z}\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-6} \lor \neg \left(z \leq 1.12 \cdot 10^{-94}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -7e+125)
     t_1
     (if (<= z -4e+54)
       (- (/ x z))
       (if (or (<= z -7.5e-6) (not (<= z 1.12e-94))) t_1 (* x (+ z 1.0)))))))

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 <= -7e+125) {
		tmp = t_1;
	} else if (z <= -4e+54) {
		tmp = -(x / z);
	} else if ((z <= -7.5e-6) || !(z <= 1.12e-94)) {
		tmp = t_1;
	} else {
		tmp = x * (z + 1.0);
	}
	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 <= (-7d+125)) then
        tmp = t_1
    else if (z <= (-4d+54)) then
        tmp = -(x / z)
    else if ((z <= (-7.5d-6)) .or. (.not. (z <= 1.12d-94))) then
        tmp = t_1
    else
        tmp = x * (z + 1.0d0)
    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 <= -7e+125) {
		tmp = t_1;
	} else if (z <= -4e+54) {
		tmp = -(x / z);
	} else if ((z <= -7.5e-6) || !(z <= 1.12e-94)) {
		tmp = t_1;
	} else {
		tmp = x * (z + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t / (b - y)
	tmp = 0
	if z <= -7e+125:
		tmp = t_1
	elif z <= -4e+54:
		tmp = -(x / z)
	elif (z <= -7.5e-6) or not (z <= 1.12e-94):
		tmp = t_1
	else:
		tmp = x * (z + 1.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -7e+125)
		tmp = t_1;
	elseif (z <= -4e+54)
		tmp = Float64(-Float64(x / z));
	elseif ((z <= -7.5e-6) || !(z <= 1.12e-94))
		tmp = t_1;
	else
		tmp = Float64(x * Float64(z + 1.0));
	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 <= -7e+125)
		tmp = t_1;
	elseif (z <= -4e+54)
		tmp = -(x / z);
	elseif ((z <= -7.5e-6) || ~((z <= 1.12e-94)))
		tmp = t_1;
	else
		tmp = x * (z + 1.0);
	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, -7e+125], t$95$1, If[LessEqual[z, -4e+54], (-N[(x / z), $MachinePrecision]), If[Or[LessEqual[z, -7.5e-6], N[Not[LessEqual[z, 1.12e-94]], $MachinePrecision]], t$95$1, N[(x * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.00000000000000023e125 or -4.0000000000000003e54 < z < -7.50000000000000019e-6 or 1.12e-94 < z

   1. Initial program 48.0%

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

   2. Taylor expanded in z around inf 72.1%

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

   3. Taylor expanded in t around inf 40.2%

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

   if -7.00000000000000023e125 < z < -4.0000000000000003e54

   1. Initial program 37.0%

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

   2. Taylor expanded in y around inf 52.2%

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

      1. +-commutative 52.2%

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

      2. mul-1-neg 52.2%

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

      3. unsub-neg 52.2%

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

   4. Simplified 52.2%

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

   5. Taylor expanded in z around inf 52.2%

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

      1. associate-*r/ 52.2%

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

      2. mul-1-neg 52.2%

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

   7. Simplified 52.2%

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

   if -7.50000000000000019e-6 < z < 1.12e-94

   1. Initial program 89.1%

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

   2. Taylor expanded in y around inf 48.8%

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

      1. +-commutative 48.8%

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

      2. mul-1-neg 48.8%

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

      3. unsub-neg 48.8%

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

   4. Simplified 48.8%

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

      1. div-inv 48.8%

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

   6. Applied egg-rr 48.8%

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

   7. Taylor expanded in z around 0 48.6%

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

      1. +-commutative 48.6%

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

   9. Simplified 48.6%

      \[\leadsto x \cdot \color{blue}{\left(z + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+125}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+54}:\\ \;\;\;\;-\frac{x}{z}\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-6} \lor \neg \left(z \leq 1.12 \cdot 10^{-94}\right):\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \end{array} \]

Alternative 19: 36.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{+133}:\\ \;\;\;\;-\frac{a}{b}\\ \mathbf{elif}\;z \leq -3 \cdot 10^{+53}:\\ \;\;\;\;-\frac{x}{z}\\ \mathbf{elif}\;z \leq -0.0033:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-94}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.85e+133)
   (- (/ a b))
   (if (<= z -3e+53)
     (- (/ x z))
     (if (<= z -0.0033) (/ t b) (if (<= z 1.2e-94) (* x (+ z 1.0)) (/ t b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.85e+133) {
		tmp = -(a / b);
	} else if (z <= -3e+53) {
		tmp = -(x / z);
	} else if (z <= -0.0033) {
		tmp = t / b;
	} else if (z <= 1.2e-94) {
		tmp = x * (z + 1.0);
	} else {
		tmp = t / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-2.85d+133)) then
        tmp = -(a / b)
    else if (z <= (-3d+53)) then
        tmp = -(x / z)
    else if (z <= (-0.0033d0)) then
        tmp = t / b
    else if (z <= 1.2d-94) then
        tmp = x * (z + 1.0d0)
    else
        tmp = t / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.85e+133) {
		tmp = -(a / b);
	} else if (z <= -3e+53) {
		tmp = -(x / z);
	} else if (z <= -0.0033) {
		tmp = t / b;
	} else if (z <= 1.2e-94) {
		tmp = x * (z + 1.0);
	} else {
		tmp = t / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -2.85e+133:
		tmp = -(a / b)
	elif z <= -3e+53:
		tmp = -(x / z)
	elif z <= -0.0033:
		tmp = t / b
	elif z <= 1.2e-94:
		tmp = x * (z + 1.0)
	else:
		tmp = t / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2.85e+133)
		tmp = Float64(-Float64(a / b));
	elseif (z <= -3e+53)
		tmp = Float64(-Float64(x / z));
	elseif (z <= -0.0033)
		tmp = Float64(t / b);
	elseif (z <= 1.2e-94)
		tmp = Float64(x * Float64(z + 1.0));
	else
		tmp = Float64(t / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -2.85e+133)
		tmp = -(a / b);
	elseif (z <= -3e+53)
		tmp = -(x / z);
	elseif (z <= -0.0033)
		tmp = t / b;
	elseif (z <= 1.2e-94)
		tmp = x * (z + 1.0);
	else
		tmp = t / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2.85e+133], (-N[(a / b), $MachinePrecision]), If[LessEqual[z, -3e+53], (-N[(x / z), $MachinePrecision]), If[LessEqual[z, -0.0033], N[(t / b), $MachinePrecision], If[LessEqual[z, 1.2e-94], N[(x * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], N[(t / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.84999999999999989e133

   1. Initial program 34.4%

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

   2. Taylor expanded in b around inf 25.5%

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

   3. Taylor expanded in t around 0 11.8%

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

      1. *-commutative 11.8%

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

      2. mul-1-neg 11.8%

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

      3. unsub-neg 11.8%

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

      4. *-commutative 11.8%

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

   5. Simplified 11.8%

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

   6. Taylor expanded in y around 0 28.2%

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

      1. associate-*r/ 28.2%

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

      2. neg-mul-1 28.2%

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

   8. Simplified 28.2%

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

   if -2.84999999999999989e133 < z < -2.99999999999999998e53

   1. Initial program 30.5%

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

   2. Taylor expanded in y around inf 47.3%

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

      1. +-commutative 47.3%

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

      2. mul-1-neg 47.3%

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

      3. unsub-neg 47.3%

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

   4. Simplified 47.3%

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

   5. Taylor expanded in z around inf 47.3%

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

      1. associate-*r/ 47.3%

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

      2. mul-1-neg 47.3%

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

   7. Simplified 47.3%

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

   if -2.99999999999999998e53 < z < -0.0033 or 1.2e-94 < z

   1. Initial program 56.7%

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

      1. fma-def 56.7%

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

      2. +-commutative 56.7%

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

      3. fma-def 56.7%

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

   3. Simplified 56.7%

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

   4. Taylor expanded in t around inf 24.5%

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

   5. Taylor expanded in b around inf 27.1%

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

   if -0.0033 < z < 1.2e-94

   1. Initial program 89.1%

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

   2. Taylor expanded in y around inf 48.8%

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

      1. +-commutative 48.8%

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

      2. mul-1-neg 48.8%

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

      3. unsub-neg 48.8%

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

   4. Simplified 48.8%

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

      1. div-inv 48.8%

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

   6. Applied egg-rr 48.8%

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

   7. Taylor expanded in z around 0 48.6%

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

      1. +-commutative 48.6%

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

   9. Simplified 48.6%

      \[\leadsto x \cdot \color{blue}{\left(z + 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{+133}:\\ \;\;\;\;-\frac{a}{b}\\ \mathbf{elif}\;z \leq -3 \cdot 10^{+53}:\\ \;\;\;\;-\frac{x}{z}\\ \mathbf{elif}\;z \leq -0.0033:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-94}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \]

Alternative 20: 52.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -3.35 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-131}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -3.35e+25)
     t_1
     (if (<= y -9.5e-31)
       (/ t (- b y))
       (if (<= y -5.5e-131)
         (* x (+ z 1.0))
         (if (<= y 1.9e+15) (/ (- t a) b) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -3.35e+25) {
		tmp = t_1;
	} else if (y <= -9.5e-31) {
		tmp = t / (b - y);
	} else if (y <= -5.5e-131) {
		tmp = x * (z + 1.0);
	} else if (y <= 1.9e+15) {
		tmp = (t - a) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (1.0d0 - z)
    if (y <= (-3.35d+25)) then
        tmp = t_1
    else if (y <= (-9.5d-31)) then
        tmp = t / (b - y)
    else if (y <= (-5.5d-131)) then
        tmp = x * (z + 1.0d0)
    else if (y <= 1.9d+15) then
        tmp = (t - a) / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -3.35e+25) {
		tmp = t_1;
	} else if (y <= -9.5e-31) {
		tmp = t / (b - y);
	} else if (y <= -5.5e-131) {
		tmp = x * (z + 1.0);
	} else if (y <= 1.9e+15) {
		tmp = (t - a) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -3.35e+25:
		tmp = t_1
	elif y <= -9.5e-31:
		tmp = t / (b - y)
	elif y <= -5.5e-131:
		tmp = x * (z + 1.0)
	elif y <= 1.9e+15:
		tmp = (t - a) / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -3.35e+25)
		tmp = t_1;
	elseif (y <= -9.5e-31)
		tmp = Float64(t / Float64(b - y));
	elseif (y <= -5.5e-131)
		tmp = Float64(x * Float64(z + 1.0));
	elseif (y <= 1.9e+15)
		tmp = Float64(Float64(t - a) / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -3.35e+25)
		tmp = t_1;
	elseif (y <= -9.5e-31)
		tmp = t / (b - y);
	elseif (y <= -5.5e-131)
		tmp = x * (z + 1.0);
	elseif (y <= 1.9e+15)
		tmp = (t - a) / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.35e+25], t$95$1, If[LessEqual[y, -9.5e-31], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.5e-131], N[(x * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e+15], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.35000000000000019e25 or 1.9e15 < y

   1. Initial program 49.2%

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

   2. Taylor expanded in y around inf 54.3%

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

      1. +-commutative 54.3%

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

      2. mul-1-neg 54.3%

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

      3. unsub-neg 54.3%

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

   4. Simplified 54.3%

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

   if -3.35000000000000019e25 < y < -9.5000000000000008e-31

   1. Initial program 45.0%

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

   2. Taylor expanded in z around inf 72.6%

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

   3. Taylor expanded in t around inf 43.0%

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

   if -9.5000000000000008e-31 < y < -5.4999999999999997e-131

   1. Initial program 86.5%

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

   2. Taylor expanded in y around inf 35.5%

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

      1. +-commutative 35.5%

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

      2. mul-1-neg 35.5%

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

      3. unsub-neg 35.5%

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

   4. Simplified 35.5%

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

      1. div-inv 35.5%

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

   6. Applied egg-rr 35.5%

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

   7. Taylor expanded in z around 0 35.9%

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

      1. +-commutative 35.9%

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

   9. Simplified 35.9%

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

   if -5.4999999999999997e-131 < y < 1.9e15

   1. Initial program 80.0%

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

   2. Taylor expanded in y around 0 56.5%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.35 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-131}:\\ \;\;\;\;x \cdot \left(z + 1\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+15}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - z}\\ \end{array} \]

Alternative 21: 36.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+133}:\\ \;\;\;\;-\frac{a}{b}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+54}:\\ \;\;\;\;-\frac{x}{z}\\ \mathbf{elif}\;z \leq -65:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.1e+133)
   (- (/ a b))
   (if (<= z -4e+54)
     (- (/ x z))
     (if (<= z -65.0) (/ t b) (if (<= z 1.6e-96) x (/ t b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.1e+133) {
		tmp = -(a / b);
	} else if (z <= -4e+54) {
		tmp = -(x / z);
	} else if (z <= -65.0) {
		tmp = t / b;
	} else if (z <= 1.6e-96) {
		tmp = x;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-3.1d+133)) then
        tmp = -(a / b)
    else if (z <= (-4d+54)) then
        tmp = -(x / z)
    else if (z <= (-65.0d0)) then
        tmp = t / b
    else if (z <= 1.6d-96) then
        tmp = x
    else
        tmp = t / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.1e+133) {
		tmp = -(a / b);
	} else if (z <= -4e+54) {
		tmp = -(x / z);
	} else if (z <= -65.0) {
		tmp = t / b;
	} else if (z <= 1.6e-96) {
		tmp = x;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -3.1e+133:
		tmp = -(a / b)
	elif z <= -4e+54:
		tmp = -(x / z)
	elif z <= -65.0:
		tmp = t / b
	elif z <= 1.6e-96:
		tmp = x
	else:
		tmp = t / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.1e+133)
		tmp = Float64(-Float64(a / b));
	elseif (z <= -4e+54)
		tmp = Float64(-Float64(x / z));
	elseif (z <= -65.0)
		tmp = Float64(t / b);
	elseif (z <= 1.6e-96)
		tmp = x;
	else
		tmp = Float64(t / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -3.1e+133)
		tmp = -(a / b);
	elseif (z <= -4e+54)
		tmp = -(x / z);
	elseif (z <= -65.0)
		tmp = t / b;
	elseif (z <= 1.6e-96)
		tmp = x;
	else
		tmp = t / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.1e+133], (-N[(a / b), $MachinePrecision]), If[LessEqual[z, -4e+54], (-N[(x / z), $MachinePrecision]), If[LessEqual[z, -65.0], N[(t / b), $MachinePrecision], If[LessEqual[z, 1.6e-96], x, N[(t / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.1e133

   1. Initial program 34.4%

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

   2. Taylor expanded in b around inf 25.5%

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

   3. Taylor expanded in t around 0 11.8%

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

      1. *-commutative 11.8%

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

      2. mul-1-neg 11.8%

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

      3. unsub-neg 11.8%

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

      4. *-commutative 11.8%

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

   5. Simplified 11.8%

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

   6. Taylor expanded in y around 0 28.2%

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

      1. associate-*r/ 28.2%

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

      2. neg-mul-1 28.2%

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

   8. Simplified 28.2%

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

   if -3.1e133 < z < -4.0000000000000003e54

   1. Initial program 30.5%

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

   2. Taylor expanded in y around inf 47.3%

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

      1. +-commutative 47.3%

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

      2. mul-1-neg 47.3%

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

      3. unsub-neg 47.3%

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

   4. Simplified 47.3%

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

   5. Taylor expanded in z around inf 47.3%

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

      1. associate-*r/ 47.3%

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

      2. mul-1-neg 47.3%

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

   7. Simplified 47.3%

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

   if -4.0000000000000003e54 < z < -65 or 1.60000000000000006e-96 < z

   1. Initial program 56.2%

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

      1. fma-def 56.2%

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

      2. +-commutative 56.2%

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

      3. fma-def 56.2%

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

   3. Simplified 56.2%

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

   4. Taylor expanded in t around inf 24.8%

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

   5. Taylor expanded in b around inf 27.4%

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

   if -65 < z < 1.60000000000000006e-96

   1. Initial program 89.2%

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

   2. Taylor expanded in z around 0 47.2%

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

4. Final simplification 37.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+133}:\\ \;\;\;\;-\frac{a}{b}\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+54}:\\ \;\;\;\;-\frac{x}{z}\\ \mathbf{elif}\;z \leq -65:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-96}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \]

Alternative 22: 63.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-6} \lor \neg \left(z \leq 2.1 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{1 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -4.6e-6) (not (<= z 2.1e-98)))
   (/ (- t a) (- b y))
   (* x (/ 1.0 (- 1.0 z)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -4.6e-6) || !(z <= 2.1e-98)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = x * (1.0 / (1.0 - z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((z <= (-4.6d-6)) .or. (.not. (z <= 2.1d-98))) then
        tmp = (t - a) / (b - y)
    else
        tmp = x * (1.0d0 / (1.0d0 - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -4.6e-6) || !(z <= 2.1e-98)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = x * (1.0 / (1.0 - z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -4.6e-6) or not (z <= 2.1e-98):
		tmp = (t - a) / (b - y)
	else:
		tmp = x * (1.0 / (1.0 - z))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -4.6e-6) || !(z <= 2.1e-98))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(x * Float64(1.0 / Float64(1.0 - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -4.6e-6) || ~((z <= 2.1e-98)))
		tmp = (t - a) / (b - y);
	else
		tmp = x * (1.0 / (1.0 - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -4.6e-6], N[Not[LessEqual[z, 2.1e-98]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.6e-6 or 2.09999999999999992e-98 < z

   1. Initial program 46.8%

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

   2. Taylor expanded in z around inf 70.1%

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

   if -4.6e-6 < z < 2.09999999999999992e-98

   1. Initial program 89.1%

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

   2. Taylor expanded in y around inf 48.8%

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

      1. +-commutative 48.8%

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

      2. mul-1-neg 48.8%

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

      3. unsub-neg 48.8%

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

   4. Simplified 48.8%

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

      1. div-inv 48.8%

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

   6. Applied egg-rr 48.8%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-6} \lor \neg \left(z \leq 2.1 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{1 - z}\\ \end{array} \]

Alternative 23: 36.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -70:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-95}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -70.0) (/ t b) (if (<= z 1.4e-95) x (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -70.0) {
		tmp = t / b;
	} else if (z <= 1.4e-95) {
		tmp = x;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-70.0d0)) then
        tmp = t / b
    else if (z <= 1.4d-95) then
        tmp = x
    else
        tmp = t / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -70.0) {
		tmp = t / b;
	} else if (z <= 1.4e-95) {
		tmp = x;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -70.0:
		tmp = t / b
	elif z <= 1.4e-95:
		tmp = x
	else:
		tmp = t / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -70.0)
		tmp = Float64(t / b);
	elseif (z <= 1.4e-95)
		tmp = x;
	else
		tmp = Float64(t / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -70.0)
		tmp = t / b;
	elseif (z <= 1.4e-95)
		tmp = x;
	else
		tmp = t / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -70.0], N[(t / b), $MachinePrecision], If[LessEqual[z, 1.4e-95], x, N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -70 or 1.4e-95 < z

   1. Initial program 46.4%

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

      1. fma-def 46.4%

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

      2. +-commutative 46.4%

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

      3. fma-def 46.5%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in t around inf 21.3%

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

   5. Taylor expanded in b around inf 25.6%

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

   if -70 < z < 1.4e-95

   1. Initial program 89.2%

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

   2. Taylor expanded in z around 0 47.2%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -70:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-95}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \]

Alternative 24: 25.9% 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 64.3%

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

2. Taylor expanded in z around 0 22.7%

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

3. Final simplification 22.7%

   \[\leadsto x \]

Developer target: 73.5% 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 2023174 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

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

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