Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, I

Average Error: 7.9 → 4.4

Time: 34.1s

Precision: binary64

Cost: 8132

Math

\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

↓

\[\begin{array}{l} t_1 := \frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y \cdot 0.5}{a}, \frac{t \cdot z}{a \cdot -0.2222222222222222}\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{y \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4.5 \cdot t}{a} \cdot z\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0))))
   (if (<= t_1 (- INFINITY))
     (fma x (/ (* y 0.5) a) (/ (* t z) (* a -0.2222222222222222)))
     (if (<= t_1 5e+303)
       (+ (* -4.5 (/ (* t z) a)) (* 0.5 (/ (* y x) a)))
       (* (/ (* -4.5 t) a) z)))))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x * y) - ((z * 9.0) * t)) / (a * 2.0);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(x, ((y * 0.5) / a), ((t * z) / (a * -0.2222222222222222)));
	} else if (t_1 <= 5e+303) {
		tmp = (-4.5 * ((t * z) / a)) + (0.5 * ((y * x) / a));
	} else {
		tmp = ((-4.5 * t) / a) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
end

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(x * y) - Float64(Float64(z * 9.0) * t)) / Float64(a * 2.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(x, Float64(Float64(y * 0.5) / a), Float64(Float64(t * z) / Float64(a * -0.2222222222222222)));
	elseif (t_1 <= 5e+303)
		tmp = Float64(Float64(-4.5 * Float64(Float64(t * z) / a)) + Float64(0.5 * Float64(Float64(y * x) / a)));
	else
		tmp = Float64(Float64(Float64(-4.5 * t) / a) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(N[(z * 9.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x * N[(N[(y * 0.5), $MachinePrecision] / a), $MachinePrecision] + N[(N[(t * z), $MachinePrecision] / N[(a * -0.2222222222222222), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+303], N[(N[(-4.5 * N[(N[(t * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(y * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-4.5 * t), $MachinePrecision] / a), $MachinePrecision] * z), $MachinePrecision]]]]

Target

Original	7.9
Target	5.7
Herbie	4.4

\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) (*.f64 a 2)) < -inf.0

   1. Initial program 64.0

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Applied egg-rr 31.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{a}, y \cdot 0.5, \frac{\left(z \cdot 9\right) \cdot t}{-2 \cdot a}\right)} \]

   3. Applied egg-rr 31.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{y \cdot 0.5}{a}, \frac{t \cdot z}{a \cdot -0.2222222222222222}\right)} \]

   if -inf.0 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) (*.f64 a 2)) < 4.9999999999999997e303

   1. Initial program 0.8

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Simplified 0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, z \cdot \left(-9 \cdot t\right)\right) \cdot \frac{0.5}{a}} \]
      Proof

   3. Taylor expanded in x around 0 0.8

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{y \cdot x}{a}} \]

   if 4.9999999999999997e303 < (/.f64 (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) (*.f64 a 2))

   1. Initial program 61.1

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2} \]

   2. Simplified 60.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, z \cdot \left(-9 \cdot t\right)\right) \cdot \frac{0.5}{a}} \]
      Proof

   3. Taylor expanded in x around 0 61.8

      \[\leadsto \color{blue}{-4.5 \cdot \frac{t \cdot z}{a}} \]

   4. Applied egg-rr 33.8

      \[\leadsto \color{blue}{\frac{-4.5 \cdot t}{a} \cdot z} \]
3. Recombined 3 regimes into one program.

Alternatives

Alternative 1
Error	6.2
Cost	1604

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -\infty:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{t \cdot z}{a} + 0.5 \cdot \frac{y \cdot x}{a}\\ \end{array} \]

Alternative 2
Error	6.2
Cost	1476

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -\infty:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(y \cdot x\right) + -4.5 \cdot \left(t \cdot z\right)}{a}\\ \end{array} \]

Alternative 3
Error	24.4
Cost	976

\[\begin{array}{l} t_1 := \left(y \cdot x\right) \cdot \frac{0.5}{a}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+40}:\\ \;\;\;\;z \cdot \left(\frac{-4.5}{a} \cdot t\right)\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-28}:\\ \;\;\;\;-4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-79}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \end{array} \]

Alternative 4
Error	24.3
Cost	976

\[\begin{array}{l} t_1 := \left(y \cdot x\right) \cdot \frac{0.5}{a}\\ t_2 := \frac{-4.5 \cdot t}{a} \cdot z\\ \mathbf{if}\;t \leq -2.12 \cdot 10^{+39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-27}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \end{array} \]

Alternative 5
Error	24.4
Cost	976

\[\begin{array}{l} t_1 := \left(y \cdot x\right) \cdot \frac{0.5}{a}\\ t_2 := \frac{-4.5 \cdot t}{a} \cdot z\\ \mathbf{if}\;t \leq -6.6 \cdot 10^{+41}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.18 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.4 \cdot 10^{-30}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-4.5 \cdot z}{a} \cdot t\\ \end{array} \]

Alternative 6
Error	24.4
Cost	976

\[\begin{array}{l} t_1 := \frac{y \cdot x}{a \cdot 2}\\ t_2 := \frac{-4.5 \cdot t}{a} \cdot z\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{+40}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.24 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.4 \cdot 10^{-27}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-4.5 \cdot z}{a} \cdot t\\ \end{array} \]

Alternative 7
Error	24.2
Cost	976

\[\begin{array}{l} t_1 := \frac{y \cdot x}{a \cdot 2}\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{+39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;t \ne 0:\\ \;\;\;\;\frac{z}{\frac{a \cdot -0.2222222222222222}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot t}{a \cdot -0.2222222222222222}\\ \end{array}\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{-4.5 \cdot t}{a} \cdot z\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-4.5 \cdot z}{a} \cdot t\\ \end{array} \]

Alternative 8
Error	8.0
Cost	832

\[-4.5 \cdot \frac{-0.1111111111111111 \cdot \left(y \cdot x\right) + t \cdot z}{a} \]

Alternative 9
Error	33.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+190}:\\ \;\;\;\;-4.5 \cdot \left(\frac{z}{a} \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{t \cdot z}{a}\\ \end{array} \]

Alternative 10
Error	33.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+191}:\\ \;\;\;\;t \cdot \left(\frac{z}{a} \cdot -4.5\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 \cdot \frac{t \cdot z}{a}\\ \end{array} \]

Alternative 11
Error	33.6
Cost	448

\[-4.5 \cdot \left(\frac{z}{a} \cdot t\right) \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))