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

Average Error: 20.3 → 4.9

Time: 10.2s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c}\\ t_2 := \frac{b}{z \cdot c}\\ t_3 := t_2 + 9 \cdot \frac{y}{\frac{z \cdot c}{x}}\\ \mathbf{if}\;t_1 \leq -1.6928749353890554 \cdot 10^{-33}:\\ \;\;\;\;t_3 - 4 \cdot \left(t \cdot \frac{a}{c}\right)\\ \mathbf{elif}\;t_1 \leq 1.920096661051147 \cdot 10^{+40}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, a \cdot -4, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_3 - 4 \cdot \left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{\sqrt[3]{a}}{\frac{\sqrt[3]{c}}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_2 + 9 \cdot \frac{y}{\frac{c}{\frac{x}{z}}}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

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)))
        (t_2 (/ b (* z c)))
        (t_3 (+ t_2 (* 9.0 (/ y (/ (* z c) x))))))
   (if (<= t_1 -1.6928749353890554e-33)
     (- t_3 (* 4.0 (* t (/ a c))))
     (if (<= t_1 1.920096661051147e+40)
       (/ (fma t (* a -4.0) (fma 9.0 (/ (* x y) z) (/ b z))) c)
       (if (<= t_1 INFINITY)
         (-
          t_3
          (*
           4.0
           (*
            (/ (* (cbrt a) (cbrt a)) (* (cbrt c) (cbrt c)))
            (/ (cbrt a) (/ (cbrt c) t)))))
         (- (+ t_2 (* 9.0 (/ y (/ c (/ x z))))) (* 4.0 (/ a (/ c t)))))))))

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 t_2 = b / (z * c);
	double t_3 = t_2 + (9.0 * (y / ((z * c) / x)));
	double tmp;
	if (t_1 <= -1.6928749353890554e-33) {
		tmp = t_3 - (4.0 * (t * (a / c)));
	} else if (t_1 <= 1.920096661051147e+40) {
		tmp = fma(t, (a * -4.0), fma(9.0, ((x * y) / z), (b / z))) / c;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_3 - (4.0 * (((cbrt(a) * cbrt(a)) / (cbrt(c) * cbrt(c))) * (cbrt(a) / (cbrt(c) / t))));
	} else {
		tmp = (t_2 + (9.0 * (y / (c / (x / z))))) - (4.0 * (a / (c / t)));
	}
	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))
	t_2 = Float64(b / Float64(z * c))
	t_3 = Float64(t_2 + Float64(9.0 * Float64(y / Float64(Float64(z * c) / x))))
	tmp = 0.0
	if (t_1 <= -1.6928749353890554e-33)
		tmp = Float64(t_3 - Float64(4.0 * Float64(t * Float64(a / c))));
	elseif (t_1 <= 1.920096661051147e+40)
		tmp = Float64(fma(t, Float64(a * -4.0), fma(9.0, Float64(Float64(x * y) / z), Float64(b / z))) / c);
	elseif (t_1 <= Inf)
		tmp = Float64(t_3 - Float64(4.0 * Float64(Float64(Float64(cbrt(a) * cbrt(a)) / Float64(cbrt(c) * cbrt(c))) * Float64(cbrt(a) / Float64(cbrt(c) / t)))));
	else
		tmp = Float64(Float64(t_2 + Float64(9.0 * Float64(y / Float64(c / Float64(x / z))))) - Float64(4.0 * Float64(a / Float64(c / t))));
	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]}, Block[{t$95$2 = N[(b / N[(z * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[(9.0 * N[(y / N[(N[(z * c), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.6928749353890554e-33], N[(t$95$3 - N[(4.0 * N[(t * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.920096661051147e+40], N[(N[(t * N[(a * -4.0), $MachinePrecision] + N[(9.0 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(t$95$3 - N[(4.0 * N[(N[(N[(N[Power[a, 1/3], $MachinePrecision] * N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] / N[(N[Power[c, 1/3], $MachinePrecision] * N[Power[c, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[a, 1/3], $MachinePrecision] / N[(N[Power[c, 1/3], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[(9.0 * N[(y / N[(c / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 * N[(a / N[(c / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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	20.3
Target	14.6
Herbie	4.9

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

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)) < -1.6928749353890554e-33

   1. Initial program 14.1

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

   2. Simplified 15.1

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

   3. Taylor expanded in t around 0 8.1

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

   4. Applied associate-/l*_binary64 7.7

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

   5. Applied associate-/l*_binary64 6.5

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

   6. Applied associate-/r/_binary64 7.0

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

   if -1.6928749353890554e-33 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < 1.92009666105114708e40

   1. Initial program 14.2

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

   2. Simplified 1.4

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

   3. Taylor expanded in x around 0 1.4

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

   4. Simplified 1.4

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

   if 1.92009666105114708e40 < (/.f64 (+.f64 (-.f64 (*.f64 (*.f64 x 9) y) (*.f64 (*.f64 (*.f64 z 4) t) a)) b) (*.f64 z c)) < +inf.0

   1. Initial program 17.2

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

   2. Simplified 17.9

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

   3. Taylor expanded in t around 0 10.7

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

   4. Applied associate-/l*_binary64 9.6

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

   5. Applied associate-/l*_binary64 7.3

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

   6. Applied *-un-lft-identity_binary64 7.3

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

   7. Applied add-cube-cbrt_binary64 7.6

      \[\leadsto \left(\frac{b}{c \cdot z} + 9 \cdot \frac{y}{\frac{c \cdot z}{x}}\right) - 4 \cdot \frac{a}{\frac{\color{blue}{\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \sqrt[3]{c}}}{1 \cdot t}} \]

   8. Applied times-frac_binary64 7.6

      \[\leadsto \left(\frac{b}{c \cdot z} + 9 \cdot \frac{y}{\frac{c \cdot z}{x}}\right) - 4 \cdot \frac{a}{\color{blue}{\frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{1} \cdot \frac{\sqrt[3]{c}}{t}}} \]

   9. Applied add-cube-cbrt_binary64 7.7

      \[\leadsto \left(\frac{b}{c \cdot z} + 9 \cdot \frac{y}{\frac{c \cdot z}{x}}\right) - 4 \cdot \frac{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{\frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{1} \cdot \frac{\sqrt[3]{c}}{t}} \]

   10. Applied times-frac_binary64 6.3

       \[\leadsto \left(\frac{b}{c \cdot z} + 9 \cdot \frac{y}{\frac{c \cdot z}{x}}\right) - 4 \cdot \color{blue}{\left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{1}} \cdot \frac{\sqrt[3]{a}}{\frac{\sqrt[3]{c}}{t}}\right)} \]

   11. Simplified 6.3

       \[\leadsto \left(\frac{b}{c \cdot z} + 9 \cdot \frac{y}{\frac{c \cdot z}{x}}\right) - 4 \cdot \left(\color{blue}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{c} \cdot \sqrt[3]{c}}} \cdot \frac{\sqrt[3]{a}}{\frac{\sqrt[3]{c}}{t}}\right) \]

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

   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 30.8

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

   3. Taylor expanded in t around 0 33.4

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

   4. Applied associate-/l*_binary64 28.7

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

   5. Applied associate-/l*_binary64 16.2

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

   6. Applied associate-/l*_binary64 6.0

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

4. Final simplification 4.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq -1.6928749353890554 \cdot 10^{-33}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{y}{\frac{z \cdot c}{x}}\right) - 4 \cdot \left(t \cdot \frac{a}{c}\right)\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq 1.920096661051147 \cdot 10^{+40}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, a \cdot -4, \mathsf{fma}\left(9, \frac{x \cdot y}{z}, \frac{b}{z}\right)\right)}{c}\\ \mathbf{elif}\;\frac{\left(\left(x \cdot 9\right) \cdot y - \left(\left(z \cdot 4\right) \cdot t\right) \cdot a\right) + b}{z \cdot c} \leq \infty:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{y}{\frac{z \cdot c}{x}}\right) - 4 \cdot \left(\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{\sqrt[3]{a}}{\frac{\sqrt[3]{c}}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{b}{z \cdot c} + 9 \cdot \frac{y}{\frac{c}{\frac{x}{z}}}\right) - 4 \cdot \frac{a}{\frac{c}{t}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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)))