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

Average Error: 20.8 → 3.8

Time: 8.4s

Precision: binary64

\[ \begin{array}{c}[t, a] = \mathsf{sort}([t, a])\\ \end{array} \]

Math

\[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

↓

\[\begin{array}{l} t_1 := \frac{b}{z \cdot c}\\ t_2 := \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(9, \frac{y}{\frac{z \cdot c}{x}}, \mathsf{fma}\left(-4, a \cdot \frac{t}{c}, t_1\right)\right)\\ \mathbf{elif}\;t_2 \leq -5 \cdot 10^{-192}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, a \cdot -4, \mathsf{fma}\left(x, 9 \cdot y, b\right) \cdot {z}^{-1}\right)}{c}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(9, \frac{y}{z} \cdot \frac{x}{c}, \mathsf{fma}\left(-4, t \cdot \frac{a}{c}, t_1\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))

↓

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ b (* z c)))
        (t_2 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))))
   (if (<= t_2 (- INFINITY))
     (fma 9.0 (/ y (/ (* z c) x)) (fma -4.0 (* a (/ t c)) t_1))
     (if (<= t_2 -5e-192)
       t_2
       (if (<= t_2 0.0)
         (/ (fma t (* a -4.0) (* (fma x (* 9.0 y) b) (pow z -1.0))) c)
         (if (<= t_2 2e+300)
           t_2
           (fma 9.0 (* (/ y z) (/ x c)) (fma -4.0 (* t (/ a c)) t_1))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
}

↓

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = b / (z * c);
	double t_2 = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma(9.0, (y / ((z * c) / x)), fma(-4.0, (a * (t / c)), t_1));
	} else if (t_2 <= -5e-192) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = fma(t, (a * -4.0), (fma(x, (9.0 * y), b) * pow(z, -1.0))) / c;
	} else if (t_2 <= 2e+300) {
		tmp = t_2;
	} else {
		tmp = fma(9.0, ((y / z) * (x / c)), fma(-4.0, (t * (a / c)), t_1));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c))
end

↓

function code(x, y, z, t, a, b, c)
	t_1 = Float64(b / Float64(z * c))
	t_2 = Float64(Float64(Float64(Float64(Float64(x * 9.0) * y) - Float64(Float64(Float64(z * 4.0) * t) * a)) + b) / Float64(z * c))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = fma(9.0, Float64(y / Float64(Float64(z * c) / x)), fma(-4.0, Float64(a * Float64(t / c)), t_1));
	elseif (t_2 <= -5e-192)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(fma(t, Float64(a * -4.0), Float64(fma(x, Float64(9.0 * y), b) * (z ^ -1.0))) / c);
	elseif (t_2 <= 2e+300)
		tmp = t_2;
	else
		tmp = fma(9.0, Float64(Float64(y / z) * Float64(x / c)), fma(-4.0, Float64(t * Float64(a / c)), t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(x * 9.0), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(z * 4.0), $MachinePrecision] * t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[(z * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(9.0 * N[(y / N[(N[(z * c), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(a * N[(t / c), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-192], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(t * N[(a * -4.0), $MachinePrecision] + N[(N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision] * N[Power[z, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$2, 2e+300], t$95$2, N[(9.0 * N[(N[(y / z), $MachinePrecision] * N[(x / c), $MachinePrecision]), $MachinePrecision] + N[(-4.0 * N[(t * N[(a / c), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	20.8
Target	14.4
Herbie	3.8

\[\begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < -1.100156740804105 \cdot 10^{-171}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 0:\\ \;\;\;\;\frac{\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z}}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 1.1708877911747488 \cdot 10^{-53}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 2.876823679546137 \cdot 10^{+130}:\\ \;\;\;\;\left(\left(9 \cdot \frac{y}{c}\right) \cdot \frac{x}{z} + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} < 1.3838515042456319 \cdot 10^{+158}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(z \cdot 4\right) \cdot \left(t \cdot a\right)\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\left(9 \cdot \left(\frac{y}{c \cdot z} \cdot x\right) + \frac{b}{c \cdot z}\right) - 4 \cdot \frac{a \cdot t}{c}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < -inf.0

   1. Initial program 64.0

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Simplified 23.7

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, a \cdot -4, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}\right)}{c}} \]

   3. Taylor expanded in t around 0 30.0

      \[\leadsto \color{blue}{\frac{b}{c \cdot z} + \left(-4 \cdot \frac{a \cdot t}{c} + 9 \cdot \frac{y \cdot x}{c \cdot z}\right)} \]

   4. Simplified 9.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(9, \frac{y}{\frac{z \cdot c}{x}}, \mathsf{fma}\left(-4, t \cdot \frac{a}{c}, \frac{b}{z \cdot c}\right)\right)} \]

   5. Taylor expanded in t around 0 15.6

      \[\leadsto \mathsf{fma}\left(9, \frac{y}{\frac{z \cdot c}{x}}, \mathsf{fma}\left(-4, \color{blue}{\frac{a \cdot t}{c}}, \frac{b}{z \cdot c}\right)\right) \]

   6. Simplified 15.3

      \[\leadsto \mathsf{fma}\left(9, \frac{y}{\frac{z \cdot c}{x}}, \mathsf{fma}\left(-4, \color{blue}{a \cdot \frac{t}{c}}, \frac{b}{z \cdot c}\right)\right) \]

   if -inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < -5.0000000000000001e-192 or -0.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < 2.0000000000000001e300

   1. Initial program 0.7

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   if -5.0000000000000001e-192 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < -0.0

   1. Initial program 31.3

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Simplified 0.7

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, a \cdot -4, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}\right)}{c}} \]

   3. Applied egg-rr 0.8

      \[\leadsto \frac{\mathsf{fma}\left(t, a \cdot -4, \color{blue}{\mathsf{fma}\left(x, 9 \cdot y, b\right) \cdot {z}^{-1}}\right)}{c} \]

   if 2.0000000000000001e300 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c))

   1. Initial program 62.3

      \[\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \]

   2. Simplified 25.5

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(t, a \cdot -4, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}\right)}{c}} \]

   3. Taylor expanded in t around 0 29.4

      \[\leadsto \color{blue}{\frac{b}{c \cdot z} + \left(-4 \cdot \frac{a \cdot t}{c} + 9 \cdot \frac{y \cdot x}{c \cdot z}\right)} \]

   4. Simplified 16.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(9, \frac{y}{\frac{z \cdot c}{x}}, \mathsf{fma}\left(-4, t \cdot \frac{a}{c}, \frac{b}{z \cdot c}\right)\right)} \]

   5. Applied egg-rr 11.3

      \[\leadsto \mathsf{fma}\left(9, \color{blue}{\frac{y}{z} \cdot \frac{x}{c}}, \mathsf{fma}\left(-4, t \cdot \frac{a}{c}, \frac{b}{z \cdot c}\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(9, \frac{y}{\frac{z \cdot c}{x}}, \mathsf{fma}\left(-4, a \cdot \frac{t}{c}, \frac{b}{z \cdot c}\right)\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -5 \cdot 10^{-192}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, a \cdot -4, \mathsf{fma}\left(x, 9 \cdot y, b\right) \cdot {z}^{-1}\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(9, \frac{y}{z} \cdot \frac{x}{c}, \mathsf{fma}\left(-4, t \cdot \frac{a}{c}, \frac{b}{z \cdot c}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022185 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, J"
  :precision binary64

  :herbie-target
  (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) -1.100156740804105e-171) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 0.0) (/ (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) z) c) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.1708877911747488e-53) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 2.876823679546137e+130) (- (+ (* (* 9.0 (/ y c)) (/ x z)) (/ b (* c z))) (* 4.0 (/ (* a t) c))) (if (< (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)) 1.3838515042456319e+158) (/ (+ (- (* (* x 9.0) y) (* (* z 4.0) (* t a))) b) (* z c)) (- (+ (* 9.0 (* (/ y (* c z)) x)) (/ b (* c z))) (* 4.0 (/ (* a t) c))))))))

  (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c)))