Development.Shake.Progress:decay from shake-0.15.5

Average Error: 22.8 → 0.7

Time: 9.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_2 := \mathsf{fma}\left(z, b - y, y\right)\\ t_3 := \frac{\mathsf{fma}\left(z, t - a, x \cdot y\right)}{t_2}\\ t_4 := \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\\ t_5 := t_4 + \frac{y}{b - y} \cdot \frac{x}{z}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+22}:\\ \;\;\;\;x \cdot \frac{y}{t_2} + t_4\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-305}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t_1 \leq 10^{-186}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_4 + \frac{x}{\frac{t_2}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

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)))))
        (t_2 (fma z (- b y) y))
        (t_3 (/ (fma z (- t a) (* x y)) t_2))
        (t_4 (/ (- t a) (+ (- b y) (/ y z))))
        (t_5 (+ t_4 (* (/ y (- b y)) (/ x z)))))
   (if (<= t_1 -1e+22)
     (+ (* x (/ y t_2)) t_4)
     (if (<= t_1 -5e-305)
       t_3
       (if (<= t_1 0.0)
         t_5
         (if (<= t_1 1e-186)
           t_3
           (if (<= t_1 INFINITY) (+ t_4 (/ x (/ t_2 y))) t_5)))))))

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 t_2 = fma(z, (b - y), y);
	double t_3 = fma(z, (t - a), (x * y)) / t_2;
	double t_4 = (t - a) / ((b - y) + (y / z));
	double t_5 = t_4 + ((y / (b - y)) * (x / z));
	double tmp;
	if (t_1 <= -1e+22) {
		tmp = (x * (y / t_2)) + t_4;
	} else if (t_1 <= -5e-305) {
		tmp = t_3;
	} else if (t_1 <= 0.0) {
		tmp = t_5;
	} else if (t_1 <= 1e-186) {
		tmp = t_3;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_4 + (x / (t_2 / y));
	} else {
		tmp = t_5;
	}
	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))))
	t_2 = fma(z, Float64(b - y), y)
	t_3 = Float64(fma(z, Float64(t - a), Float64(x * y)) / t_2)
	t_4 = Float64(Float64(t - a) / Float64(Float64(b - y) + Float64(y / z)))
	t_5 = Float64(t_4 + Float64(Float64(y / Float64(b - y)) * Float64(x / z)))
	tmp = 0.0
	if (t_1 <= -1e+22)
		tmp = Float64(Float64(x * Float64(y / t_2)) + t_4);
	elseif (t_1 <= -5e-305)
		tmp = t_3;
	elseif (t_1 <= 0.0)
		tmp = t_5;
	elseif (t_1 <= 1e-186)
		tmp = t_3;
	elseif (t_1 <= Inf)
		tmp = Float64(t_4 + Float64(x / Float64(t_2 / y)));
	else
		tmp = t_5;
	end
	return 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]}, Block[{t$95$2 = N[(z * N[(b - y), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z * N[(t - a), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t - a), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 + N[(N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+22], N[(N[(x * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$1, -5e-305], t$95$3, If[LessEqual[t$95$1, 0.0], t$95$5, If[LessEqual[t$95$1, 1e-186], t$95$3, If[LessEqual[t$95$1, Infinity], N[(t$95$4 + N[(x / N[(t$95$2 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$5]]]]]]]]]]

Target

Original	22.8
Target	17.7
Herbie	0.7

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1e22

   1. Initial program 21.3

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

   2. Simplified 21.2

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

   3. Taylor expanded in t around 0 21.2

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

   4. Simplified 7.5

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

   5. Applied egg-rr 0.1

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

   6. Taylor expanded in z around 0 0.1

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

   7. Simplified 0.1

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

   if -1e22 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -4.99999999999999985e-305 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 9.9999999999999991e-187

   1. Initial program 0.4

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

   2. Simplified 0.4

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

   if -4.99999999999999985e-305 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 57.3

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

   2. Simplified 57.3

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

   3. Taylor expanded in t around 0 57.3

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

   4. Simplified 54.5

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

   5. Applied egg-rr 54.5

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

   6. Taylor expanded in z around 0 12.2

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

   7. Simplified 12.2

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

   8. Taylor expanded in z around inf 17.4

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

   9. Simplified 0.8

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

   if 9.9999999999999991e-187 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 13.7

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

   2. Simplified 13.7

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

   3. Taylor expanded in t around 0 13.7

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

   4. Simplified 5.5

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

   5. Applied egg-rr 1.3

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

   6. Taylor expanded in z around 0 1.2

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

   7. Simplified 1.2

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

   8. Applied egg-rr 1.2

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -1 \cdot 10^{+22}:\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(z, b - y, y\right)} + \frac{t - a}{\left(b - y\right) + \frac{y}{z}}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 0:\\ \;\;\;\;\frac{t - a}{\left(b - y\right) + \frac{y}{z}} + \frac{y}{b - y} \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq 10^{-186}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, t - a, x \cdot y\right)}{\mathsf{fma}\left(z, b - y, y\right)}\\ \mathbf{elif}\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \leq \infty:\\ \;\;\;\;\frac{t - a}{\left(b - y\right) + \frac{y}{z}} + \frac{x}{\frac{\mathsf{fma}\left(z, b - y, y\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{\left(b - y\right) + \frac{y}{z}} + \frac{y}{b - y} \cdot \frac{x}{z}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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)))))