Development.Shake.Progress:decay from shake-0.15.5

Average Accuracy: 63.7% → 91.5%

Time: 30.8s

Precision: binary64

Cost: 13201

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{a - t}{\frac{z + -1}{z}} - \frac{b}{\frac{{\left(z + -1\right)}^{2}}{x \cdot z}}}{y} - \frac{x}{z + -1}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-298} \lor \neg \left(t_1 \leq 0\right) \land t_1 \leq 10^{+288}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{b - y} + \left(\frac{t}{b - y} - \left(\frac{a}{b - y} + \frac{y}{z} \cdot \frac{t - a}{{\left(b - y\right)}^{2}}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))))
   (if (<= t_1 (- INFINITY))
     (-
      (/
       (- (/ (- a t) (/ (+ z -1.0) z)) (/ b (/ (pow (+ z -1.0) 2.0) (* x z))))
       y)
      (/ x (+ z -1.0)))
     (if (or (<= t_1 -2e-298) (and (not (<= t_1 0.0)) (<= t_1 1e+288)))
       t_1
       (+
        (* (/ y z) (/ x (- b y)))
        (-
         (/ t (- b y))
         (+ (/ a (- b y)) (* (/ y z) (/ (- t a) (pow (- b y) 2.0))))))))))

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)));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = ((((a - t) / ((z + -1.0) / z)) - (b / (pow((z + -1.0), 2.0) / (x * z)))) / y) - (x / (z + -1.0));
	} else if ((t_1 <= -2e-298) || (!(t_1 <= 0.0) && (t_1 <= 1e+288))) {
		tmp = t_1;
	} else {
		tmp = ((y / z) * (x / (b - y))) + ((t / (b - y)) - ((a / (b - y)) + ((y / z) * ((t - a) / pow((b - y), 2.0)))));
	}
	return tmp;
}

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)));
}

↓

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = ((((a - t) / ((z + -1.0) / z)) - (b / (Math.pow((z + -1.0), 2.0) / (x * z)))) / y) - (x / (z + -1.0));
	} else if ((t_1 <= -2e-298) || (!(t_1 <= 0.0) && (t_1 <= 1e+288))) {
		tmp = t_1;
	} else {
		tmp = ((y / z) * (x / (b - y))) + ((t / (b - y)) - ((a / (b - y)) + ((y / z) * ((t - a) / Math.pow((b - y), 2.0)))));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a, b):
	t_1 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = ((((a - t) / ((z + -1.0) / z)) - (b / (math.pow((z + -1.0), 2.0) / (x * z)))) / y) - (x / (z + -1.0))
	elif (t_1 <= -2e-298) or (not (t_1 <= 0.0) and (t_1 <= 1e+288)):
		tmp = t_1
	else:
		tmp = ((y / z) * (x / (b - y))) + ((t / (b - y)) - ((a / (b - y)) + ((y / z) * ((t - a) / math.pow((b - y), 2.0)))))
	return tmp

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

↓

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(Float64(a - t) / Float64(Float64(z + -1.0) / z)) - Float64(b / Float64((Float64(z + -1.0) ^ 2.0) / Float64(x * z)))) / y) - Float64(x / Float64(z + -1.0)));
	elseif ((t_1 <= -2e-298) || (!(t_1 <= 0.0) && (t_1 <= 1e+288)))
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(y / z) * Float64(x / Float64(b - y))) + Float64(Float64(t / Float64(b - y)) - Float64(Float64(a / Float64(b - y)) + Float64(Float64(y / z) * Float64(Float64(t - a) / (Float64(b - y) ^ 2.0))))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = ((((a - t) / ((z + -1.0) / z)) - (b / (((z + -1.0) ^ 2.0) / (x * z)))) / y) - (x / (z + -1.0));
	elseif ((t_1 <= -2e-298) || (~((t_1 <= 0.0)) && (t_1 <= 1e+288)))
		tmp = t_1;
	else
		tmp = ((y / z) * (x / (b - y))) + ((t / (b - y)) - ((a / (b - y)) + ((y / z) * ((t - a) / ((b - y) ^ 2.0)))));
	end
	tmp_2 = tmp;
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]

↓

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(N[(a - t), $MachinePrecision] / N[(N[(z + -1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - N[(b / N[(N[Power[N[(z + -1.0), $MachinePrecision], 2.0], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - N[(x / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$1, -2e-298], And[N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision], LessEqual[t$95$1, 1e+288]]], t$95$1, N[(N[(N[(y / z), $MachinePrecision] * N[(x / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(N[(y / z), $MachinePrecision] * N[(N[(t - a), $MachinePrecision] / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	63.7%
Target	71.9%
Herbie	91.5%

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 0.0%

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

   2. Taylor expanded in y around -inf 35.2%

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

   3. Simplified 59.7%

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

      [Start] 35.2	\[ -1 \cdot \frac{x}{z - 1} + -1 \cdot \frac{\frac{\left(t - a\right) \cdot z}{z - 1} - -1 \cdot \frac{z \cdot \left(b \cdot x\right)}{{\left(z - 1\right)}^{2}}}{y} \]
      distribute-lft-out [=>] 35.2	\[ \color{blue}{-1 \cdot \left(\frac{x}{z - 1} + \frac{\frac{\left(t - a\right) \cdot z}{z - 1} - -1 \cdot \frac{z \cdot \left(b \cdot x\right)}{{\left(z - 1\right)}^{2}}}{y}\right)} \]

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1.99999999999999982e-298 or -0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1e288

   1. Initial program 99.5%

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

   if -1.99999999999999982e-298 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -0.0 or 1e288 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 9.2%

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

   2. Taylor expanded in z around inf 46.1%

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

   3. Simplified 83.9%

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

      [Start] 46.1	\[ \left(\frac{y \cdot x}{z \cdot \left(b - y\right)} + \frac{t}{b - y}\right) - \left(\frac{\left(t - a\right) \cdot y}{z \cdot {\left(b - y\right)}^{2}} + \frac{a}{b - y}\right) \]
      associate--l+ [=>] 46.1	\[ \color{blue}{\frac{y \cdot x}{z \cdot \left(b - y\right)} + \left(\frac{t}{b - y} - \left(\frac{\left(t - a\right) \cdot y}{z \cdot {\left(b - y\right)}^{2}} + \frac{a}{b - y}\right)\right)} \]
      times-frac [=>] 58.8	\[ \color{blue}{\frac{y}{z} \cdot \frac{x}{b - y}} + \left(\frac{t}{b - y} - \left(\frac{\left(t - a\right) \cdot y}{z \cdot {\left(b - y\right)}^{2}} + \frac{a}{b - y}\right)\right) \]
      +-commutative [=>] 58.8	\[ \frac{y}{z} \cdot \frac{x}{b - y} + \left(\frac{t}{b - y} - \color{blue}{\left(\frac{a}{b - y} + \frac{\left(t - a\right) \cdot y}{z \cdot {\left(b - y\right)}^{2}}\right)}\right) \]
      *-commutative [<=] 58.8	\[ \frac{y}{z} \cdot \frac{x}{b - y} + \left(\frac{t}{b - y} - \left(\frac{a}{b - y} + \frac{\left(t - a\right) \cdot y}{\color{blue}{{\left(b - y\right)}^{2} \cdot z}}\right)\right) \]
      times-frac [=>] 83.9	\[ \frac{y}{z} \cdot \frac{x}{b - y} + \left(\frac{t}{b - y} - \left(\frac{a}{b - y} + \color{blue}{\frac{t - a}{{\left(b - y\right)}^{2}} \cdot \frac{y}{z}}\right)\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -\infty:\\ \;\;\;\;\frac{\frac{a - t}{\frac{z + -1}{z}} - \frac{b}{\frac{{\left(z + -1\right)}^{2}}{x \cdot z}}}{y} - \frac{x}{z + -1}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -2 \cdot 10^{-298} \lor \neg \left(\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0\right) \land \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 10^{+288}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{b - y} + \left(\frac{t}{b - y} - \left(\frac{a}{b - y} + \frac{y}{z} \cdot \frac{t - a}{{\left(b - y\right)}^{2}}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.6%
Cost	9348

\[\begin{array}{l} t_1 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{\frac{a - t}{\frac{z + -1}{z}} - \frac{b}{\frac{{\left(z + -1\right)}^{2}}{x \cdot z}}}{y} - \frac{x}{z + -1}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-298} \lor \neg \left(t_1 \leq 0\right) \land t_1 \leq 10^{+288}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]

Alternative 2
Accuracy	86.1%
Cost	5713

\[\begin{array}{l} t_1 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{-x}{z + -1}\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{-298} \lor \neg \left(t_1 \leq 0\right) \land t_1 \leq 10^{+288}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]

Alternative 3
Accuracy	73.2%
Cost	1228

\[\begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.22 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-66}:\\ \;\;\;\;x + \frac{t - a}{\frac{y}{z}}\\ \mathbf{elif}\;z \leq 0.0155:\\ \;\;\;\;\frac{x \cdot y + z \cdot t}{y + z \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	81.2%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;z \leq -3900 \lor \neg \left(z \leq 48000000000000\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot b}\\ \end{array} \]

Alternative 5
Accuracy	71.0%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-26} \lor \neg \left(z \leq 2.7 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{y}{t - a}}\\ \end{array} \]

Alternative 6
Accuracy	72.7%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-26} \lor \neg \left(z \leq 2.1 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t - a}{\frac{y}{z}}\\ \end{array} \]

Alternative 7
Accuracy	51.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-26} \lor \neg \left(z \leq 2.8 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{t}{y}\\ \end{array} \]

Alternative 8
Accuracy	53.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-26} \lor \neg \left(z \leq 3.6 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot t}{y}\\ \end{array} \]

Alternative 9
Accuracy	69.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{-26} \lor \neg \left(z \leq 2.5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot t}{y}\\ \end{array} \]

Alternative 10
Accuracy	53.1%
Cost	649

\[\begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-11} \lor \neg \left(y \leq 1.6 \cdot 10^{+52}\right):\\ \;\;\;\;\frac{-x}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \]

Alternative 11
Accuracy	45.3%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -0.00031 \lor \neg \left(z \leq 3.1 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot z\\ \end{array} \]

Alternative 12
Accuracy	47.7%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-35} \lor \neg \left(z \leq 7.2 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Accuracy	37.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -0.00016:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-33}:\\ \;\;\;\;x + x \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \]

Alternative 14
Accuracy	34.5%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1150:\\ \;\;\;\;\frac{a}{y}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{y}\\ \end{array} \]

Alternative 15
Accuracy	37.3%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -255000:\\ \;\;\;\;\frac{t}{b}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-34}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \]

Alternative 16
Accuracy	26.9%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023137 
(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)))))