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

Average Error: 3.7 → 0.4

Time: 12.6s

Precision: binary64

Cost: 1480

Math

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

↓

\[\begin{array}{l} t_1 := \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{if}\;z \cdot 3 \leq -4 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot 3 \leq 10:\\ \;\;\;\;x + \frac{1}{z} \cdot \frac{y - \frac{t}{y}}{-3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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) -4e-37)
     t_1
     (if (<= (* z 3.0) 10.0) (+ x (* (/ 1.0 z) (/ (- y (/ t y)) -3.0))) 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) <= -4e-37) {
		tmp = t_1;
	} else if ((z * 3.0) <= 10.0) {
		tmp = x + ((1.0 / z) * ((y - (t / y)) / -3.0));
	} 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) <= (-4d-37)) then
        tmp = t_1
    else if ((z * 3.0d0) <= 10.0d0) then
        tmp = x + ((1.0d0 / z) * ((y - (t / y)) / (-3.0d0)))
    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) <= -4e-37) {
		tmp = t_1;
	} else if ((z * 3.0) <= 10.0) {
		tmp = x + ((1.0 / z) * ((y - (t / y)) / -3.0));
	} 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) <= -4e-37:
		tmp = t_1
	elif (z * 3.0) <= 10.0:
		tmp = x + ((1.0 / z) * ((y - (t / y)) / -3.0))
	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) <= -4e-37)
		tmp = t_1;
	elseif (Float64(z * 3.0) <= 10.0)
		tmp = Float64(x + Float64(Float64(1.0 / z) * Float64(Float64(y - Float64(t / y)) / -3.0)));
	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) <= -4e-37)
		tmp = t_1;
	elseif ((z * 3.0) <= 10.0)
		tmp = x + ((1.0 / z) * ((y - (t / y)) / -3.0));
	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], -4e-37], t$95$1, If[LessEqual[N[(z * 3.0), $MachinePrecision], 10.0], N[(x + N[(N[(1.0 / z), $MachinePrecision] * N[(N[(y - N[(t / y), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	3.7
Target	1.8
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z 3) < -4.00000000000000027e-37 or 10 < (*.f64 z 3)

   1. Initial program 0.4

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

   if -4.00000000000000027e-37 < (*.f64 z 3) < 10

   1. Initial program 11.4

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

   2. Simplified 0.3

      \[\leadsto \color{blue}{x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)} \]
      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)))): 24 points increase in error, 23 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))))): 0 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)))): 11 points increase in error, 18 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))))): 25 points increase in error, 19 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. Applied egg-rr 0.3

      \[\leadsto x + \color{blue}{\frac{y - \frac{t}{y}}{z \cdot -3}} \]

   4. Applied egg-rr 0.3

      \[\leadsto x + \color{blue}{\frac{1}{z} \cdot \frac{y - \frac{t}{y}}{-3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \leq -4 \cdot 10^{-37}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;z \cdot 3 \leq 10:\\ \;\;\;\;x + \frac{1}{z} \cdot \frac{y - \frac{t}{y}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	1480

\[\begin{array}{l} t_1 := \frac{t}{\left(z \cdot 3\right) \cdot y} + \left(x + -0.3333333333333333 \cdot \frac{y}{z}\right)\\ \mathbf{if}\;z \cdot 3 \leq -1 \cdot 10^{+55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot 3 \leq 5 \cdot 10^{-114}:\\ \;\;\;\;x + \frac{y - \frac{t}{y}}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	32.2
Cost	1112

\[\begin{array}{l} t_1 := 0.3333333333333333 \cdot \frac{t}{z \cdot y}\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{+100}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-212}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 10^{-168}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 21:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \end{array} \]

Alternative 3
Error	31.1
Cost	1108

\[\begin{array}{l} t_1 := 0.3333333333333333 \cdot \frac{\frac{t}{z}}{y}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+100}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq -1.38 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -1.12 \cdot 10^{-99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5.4 \cdot 10^{-210}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \end{array} \]

Alternative 4
Error	9.2
Cost	1104

\[\begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{-7}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-114}:\\ \;\;\;\;x + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-77}:\\ \;\;\;\;\left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-37}:\\ \;\;\;\;x + \frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \]

Alternative 5
Error	1.5
Cost	1092

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+69}:\\ \;\;\;\;x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{0.3333333333333333}{y} \cdot \frac{t}{z} + \frac{\frac{y}{-3}}{z}\right)\\ \end{array} \]

Alternative 6
Error	16.3
Cost	976

\[\begin{array}{l} t_1 := x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{if}\;y \leq -3.4 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-94}:\\ \;\;\;\;\frac{0.3333333333333333}{y} \cdot \frac{t}{z}\\ \mathbf{elif}\;y \leq -2.65 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-164}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	16.3
Cost	976

\[\begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -3.4 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-94}:\\ \;\;\;\;\frac{0.3333333333333333}{y} \cdot \frac{t}{z}\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-215}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.35 \cdot 10^{-162}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 8
Error	16.2
Cost	976

\[\begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{y}}{\frac{z}{t}}\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-210}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-164}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\frac{t}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 9
Error	16.3
Cost	976

\[\begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -3.4 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{y}}{\frac{z}{t}}\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-212}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{t}{z}}{3 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 10
Error	16.3
Cost	976

\[\begin{array}{l} t_1 := x + \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{y}}{\frac{z}{t}}\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{-212}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.1 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{\frac{t}{z}}{y}}{3}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \end{array} \]

Alternative 11
Error	1.8
Cost	968

\[\begin{array}{l} t_1 := x + \left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{if}\;y \leq -4 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-112}:\\ \;\;\;\;x + \frac{\frac{t}{z}}{3 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	1.8
Cost	968

\[\begin{array}{l} t_1 := x + \frac{y - \frac{t}{y}}{z \cdot -3}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{-119}:\\ \;\;\;\;x + \frac{\frac{t}{z}}{3 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	11.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -1.52 \cdot 10^{-117}:\\ \;\;\;\;x + \frac{-0.3333333333333333}{\frac{z}{y}}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+64}:\\ \;\;\;\;\left(y - \frac{t}{y}\right) \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \]

Alternative 14
Error	8.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -2.75 \cdot 10^{-8}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-35}:\\ \;\;\;\;x + \frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \]

Alternative 15
Error	5.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-7}:\\ \;\;\;\;x + y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-33}:\\ \;\;\;\;x + \frac{\frac{t}{z}}{3 \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z \cdot -3}\\ \end{array} \]

Alternative 16
Error	28.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+40}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+65}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 17
Error	27.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+40}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+64}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 18
Error	37.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022330 
(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))))