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

Average Error: 20.1 → 8.0

Time: 13.5s

Precision: binary64

\[[x, y] = \mathsf{sort}([x, y]) \\]

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;{\left(c \cdot \frac{1}{\mathsf{fma}\left(t, a \cdot -4, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}\right)}\right)}^{-1}\\ \mathbf{elif}\;t_1 \leq -2.294825344272469 \cdot 10^{-242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 3.409990370591344 \cdot 10^{-202}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, a \cdot -4, \frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right)}{c}\\ \mathbf{elif}\;t_1 \leq 5.417585858280621 \cdot 10^{+285}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, -4 \cdot \left(t \cdot a\right)\right)}{c}\\ \end{array} \]

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 (/ (+ (- (* (* x 9.0) y) (* (* (* z 4.0) t) a)) b) (* z c))))
   (if (<= t_1 (- INFINITY))
     (pow (* c (/ 1.0 (fma t (* a -4.0) (/ (fma x (* 9.0 y) b) z)))) -1.0)
     (if (<= t_1 -2.294825344272469e-242)
       t_1
       (if (<= t_1 3.409990370591344e-202)
         (/ (fma t (* a -4.0) (/ (+ b (* 9.0 (* x y))) z)) c)
         (if (<= t_1 5.417585858280621e+285)
           t_1
           (/ (fma 9.0 (/ (* x y) z) (* -4.0 (* t a))) c)))))))

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 = ((((x * 9.0) * y) - (((z * 4.0) * t) * a)) + b) / (z * c);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = pow((c * (1.0 / fma(t, (a * -4.0), (fma(x, (9.0 * y), b) / z)))), -1.0);
	} else if (t_1 <= -2.294825344272469e-242) {
		tmp = t_1;
	} else if (t_1 <= 3.409990370591344e-202) {
		tmp = fma(t, (a * -4.0), ((b + (9.0 * (x * y))) / z)) / c;
	} else if (t_1 <= 5.417585858280621e+285) {
		tmp = t_1;
	} else {
		tmp = fma(9.0, ((x * y) / z), (-4.0 * (t * a))) / c;
	}
	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(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_1 <= Float64(-Inf))
		tmp = Float64(c * Float64(1.0 / fma(t, Float64(a * -4.0), Float64(fma(x, Float64(9.0 * y), b) / z)))) ^ -1.0;
	elseif (t_1 <= -2.294825344272469e-242)
		tmp = t_1;
	elseif (t_1 <= 3.409990370591344e-202)
		tmp = Float64(fma(t, Float64(a * -4.0), Float64(Float64(b + Float64(9.0 * Float64(x * y))) / z)) / c);
	elseif (t_1 <= 5.417585858280621e+285)
		tmp = t_1;
	else
		tmp = Float64(fma(9.0, Float64(Float64(x * y) / z), Float64(-4.0 * Float64(t * a))) / c);
	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[(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$1, (-Infinity)], N[Power[N[(c * N[(1.0 / N[(t * N[(a * -4.0), $MachinePrecision] + N[(N[(x * N[(9.0 * y), $MachinePrecision] + b), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[t$95$1, -2.294825344272469e-242], t$95$1, If[LessEqual[t$95$1, 3.409990370591344e-202], N[(N[(t * N[(a * -4.0), $MachinePrecision] + N[(N[(b + N[(9.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, 5.417585858280621e+285], t$95$1, N[(N[(9.0 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision] + N[(-4.0 * N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	20.1
Target	14.4
Herbie	8.0

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

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 25.8

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

   3. Applied egg-rr 25.8

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

   4. Applied egg-rr 25.9

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

   if -inf.0 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < -2.294825344272469e-242 or 3.4099903705913438e-202 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < 5.41758585828062105e285

   1. Initial program 0.6

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

   if -2.294825344272469e-242 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < 3.4099903705913438e-202

   1. Initial program 31.6

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

   2. Simplified 1.1

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

   3. Applied egg-rr 1.2

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

   4. Taylor expanded in z around 0 1.1

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

   if 5.41758585828062105e285 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c))

   1. Initial program 59.6

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

   2. Simplified 26.3

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

   3. Applied egg-rr 26.5

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

   4. Taylor expanded in b around 0 30.5

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

   5. Simplified 30.5

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

4. Final simplification 8.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -\infty:\\ \;\;\;\;{\left(c \cdot \frac{1}{\mathsf{fma}\left(t, a \cdot -4, \frac{\mathsf{fma}\left(x, 9 \cdot y, b\right)}{z}\right)}\right)}^{-1}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -2.294825344272469 \cdot 10^{-242}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 3.409990370591344 \cdot 10^{-202}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, a \cdot -4, \frac{b + 9 \cdot \left(x \cdot y\right)}{z}\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 5.417585858280621 \cdot 10^{+285}:\\ \;\;\;\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(9, \frac{x \cdot y}{z}, -4 \cdot \left(t \cdot a\right)\right)}{c}\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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)))