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

Average Error: 3.5 → 0.6

Time: 10.9s

Precision: binary64

Cost: 1480

Math

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

↓

\[\begin{array}{l} t_1 := \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{+65}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y)))))
   (if (<= (* z 3.0) -1e+31)
     t_1
     (if (<= (* z 3.0) 5e+65)
       (+ x (* (/ -0.3333333333333333 z) (- y (/ t y))))
       t_1))))

C

double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	double tmp;
	if ((z * 3.0) <= -1e+31) {
		tmp = t_1;
	} else if ((z * 3.0) <= 5e+65) {
		tmp = x + ((-0.3333333333333333 / z) * (y - (t / y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - (y / (z * 3.0d0))) + (t / ((z * 3.0d0) * y))
    if ((z * 3.0d0) <= (-1d+31)) then
        tmp = t_1
    else if ((z * 3.0d0) <= 5d+65) then
        tmp = x + (((-0.3333333333333333d0) / z) * (y - (t / y)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
}

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	double tmp;
	if ((z * 3.0) <= -1e+31) {
		tmp = t_1;
	} else if ((z * 3.0) <= 5e+65) {
		tmp = x + ((-0.3333333333333333 / z) * (y - (t / y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))

↓

def code(x, y, z, t):
	t_1 = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y))
	tmp = 0
	if (z * 3.0) <= -1e+31:
		tmp = t_1
	elif (z * 3.0) <= 5e+65:
		tmp = x + ((-0.3333333333333333 / z) * (y - (t / y)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	return Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
end

↓

function code(x, y, z, t)
	t_1 = Float64(Float64(x - Float64(y / Float64(z * 3.0))) + Float64(t / Float64(Float64(z * 3.0) * y)))
	tmp = 0.0
	if (Float64(z * 3.0) <= -1e+31)
		tmp = t_1;
	elseif (Float64(z * 3.0) <= 5e+65)
		tmp = Float64(x + Float64(Float64(-0.3333333333333333 / z) * Float64(y - Float64(t / y))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
end

↓

function tmp_2 = code(x, y, z, t)
	t_1 = (x - (y / (z * 3.0))) + (t / ((z * 3.0) * y));
	tmp = 0.0;
	if ((z * 3.0) <= -1e+31)
		tmp = t_1;
	elseif ((z * 3.0) <= 5e+65)
		tmp = x + ((-0.3333333333333333 / z) * (y - (t / y)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(z * 3.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * 3.0), $MachinePrecision], -1e+31], t$95$1, If[LessEqual[N[(z * 3.0), $MachinePrecision], 5e+65], N[(x + N[(N[(-0.3333333333333333 / z), $MachinePrecision] * N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	3.5
Target	1.7
Herbie	0.6

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y} \]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z 3) < -9.9999999999999996e30 or 4.99999999999999973e65 < (*.f64 z 3)

   1. Initial program 0.5

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   if -9.9999999999999996e30 < (*.f64 z 3) < 4.99999999999999973e65

   1. Initial program 7.6

      \[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y} \]

   2. Simplified 0.6

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
      Proof
      (+.f64 x (*.f64 (/.f64 -1/3 z) (-.f64 y (/.f64 t y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (*.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 -1 3)) z) (-.f64 y (/.f64 t y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 -1 (*.f64 3 z))) (-.f64 y (/.f64 t y)))): 31 points increase in error, 16 points decrease in error
      (+.f64 x (*.f64 (/.f64 -1 (Rewrite<= *-commutative_binary64 (*.f64 z 3))) (-.f64 y (/.f64 t y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 -1 (*.f64 z 3)) y) (*.f64 (/.f64 -1 (*.f64 z 3)) (/.f64 t y))))): 1 points increase in error, 1 points decrease in error
      (+.f64 x (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 -1 y) (*.f64 z 3))) (*.f64 (/.f64 -1 (*.f64 z 3)) (/.f64 t y)))): 6 points increase in error, 20 points decrease in error
      (+.f64 x (-.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 y)) (*.f64 z 3)) (*.f64 (/.f64 -1 (*.f64 z 3)) (/.f64 t y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 y (*.f64 z 3)))) (*.f64 (/.f64 -1 (*.f64 z 3)) (/.f64 t y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (neg.f64 (/.f64 y (*.f64 z 3))) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 t) (*.f64 (*.f64 z 3) y))))): 21 points increase in error, 28 points decrease in error
      (+.f64 x (-.f64 (neg.f64 (/.f64 y (*.f64 z 3))) (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 t)) (*.f64 (*.f64 z 3) y)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (-.f64 (neg.f64 (/.f64 y (*.f64 z 3))) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 t (*.f64 (*.f64 z 3) y)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= unsub-neg_binary64 (+.f64 (neg.f64 (/.f64 y (*.f64 z 3))) (neg.f64 (neg.f64 (/.f64 t (*.f64 (*.f64 z 3) y))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (/.f64 y (*.f64 z 3)) (neg.f64 (/.f64 t (*.f64 (*.f64 z 3) y))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 y (*.f64 z 3)) (/.f64 t (*.f64 (*.f64 z 3) y)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 x (-.f64 (/.f64 y (*.f64 z 3)) (/.f64 t (*.f64 (*.f64 z 3) y))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 x (/.f64 y (*.f64 z 3))) (/.f64 t (*.f64 (*.f64 z 3) y)))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+31}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{+65}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	1480

\[\begin{array}{l} t_1 := \frac{t}{\left(z \cdot 3\right) \cdot y} + \left(x + -0.3333333333333333 \cdot \frac{y}{z}\right)\\ \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{+156}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	30.8
Cost	1376

\[\begin{array}{l} t_1 := \frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{if}\;y \leq -4.9 \cdot 10^{+86}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-133}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -3.25 \cdot 10^{-170}:\\ \;\;\;\;t \cdot \frac{\frac{0.3333333333333333}{y}}{z}\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{-241}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-245}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-122}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{z \cdot y}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{+77}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \end{array} \]

Alternative 3
Error	1.8
Cost	968

\[\begin{array}{l} t_1 := x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)\\ \mathbf{if}\;y \leq -4.3 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-118}:\\ \;\;\;\;x + \frac{\frac{t}{z}}{3 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	1.7
Cost	968

\[\begin{array}{l} t_1 := y - \frac{t}{y}\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{-101}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{z} \cdot t_1\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{-84}:\\ \;\;\;\;x + \frac{\frac{t}{z}}{3 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t_1}{z \cdot -3}\\ \end{array} \]

Alternative 5
Error	28.0
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+45}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{-142}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-47}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{z \cdot y}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	29.1
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+88}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-128}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-103}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+77}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \end{array} \]

Alternative 7
Error	11.3
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+35}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-49}:\\ \;\;\;\;\frac{0.3333333333333333}{z} \cdot \left(\frac{t}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;x + -0.3333333333333333 \cdot \frac{y}{z}\\ \end{array} \]

Alternative 8
Error	6.7
Cost	840

\[\begin{array}{l} t_1 := x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{if}\;y \leq -4.2 \cdot 10^{+60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-40}:\\ \;\;\;\;x + \frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	15.7
Cost	712

\[\begin{array}{l} t_1 := x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{if}\;y \leq -1.9 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-103}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	15.8
Cost	712

\[\begin{array}{l} t_1 := x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{\frac{z}{t}}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	27.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+43}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	27.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+43}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	38.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022331 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :herbie-target
  (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y))

  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))