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

Average Error: 3.6 → 1.0

Time: 18.0s

Precision: binary64

Cost: 1352

Math

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

↓

\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;t \leq -6.067626527897145 \cdot 10^{-58}:\\ \;\;\;\;t_1 + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{elif}\;t \leq 3.830453512798634 \cdot 10^{-203}:\\ \;\;\;\;\left(x + \frac{y}{3} \cdot \frac{-1}{z}\right) + \frac{\frac{t}{y}}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{0.3333333333333333}{y \cdot \frac{z}{t}}\\ \end{array} \]

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)))))
   (if (<= t -6.067626527897145e-58)
     (+ t_1 (/ t (* z (* y 3.0))))
     (if (<= t 3.830453512798634e-203)
       (+ (+ x (* (/ y 3.0) (/ -1.0 z))) (/ (/ t y) (* z 3.0)))
       (+ t_1 (/ 0.3333333333333333 (* y (/ z t))))))))

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));
	double tmp;
	if (t <= -6.067626527897145e-58) {
		tmp = t_1 + (t / (z * (y * 3.0)));
	} else if (t <= 3.830453512798634e-203) {
		tmp = (x + ((y / 3.0) * (-1.0 / z))) + ((t / y) / (z * 3.0));
	} else {
		tmp = t_1 + (0.3333333333333333 / (y * (z / t)));
	}
	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))
    if (t <= (-6.067626527897145d-58)) then
        tmp = t_1 + (t / (z * (y * 3.0d0)))
    else if (t <= 3.830453512798634d-203) then
        tmp = (x + ((y / 3.0d0) * ((-1.0d0) / z))) + ((t / y) / (z * 3.0d0))
    else
        tmp = t_1 + (0.3333333333333333d0 / (y * (z / t)))
    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));
	double tmp;
	if (t <= -6.067626527897145e-58) {
		tmp = t_1 + (t / (z * (y * 3.0)));
	} else if (t <= 3.830453512798634e-203) {
		tmp = (x + ((y / 3.0) * (-1.0 / z))) + ((t / y) / (z * 3.0));
	} else {
		tmp = t_1 + (0.3333333333333333 / (y * (z / t)));
	}
	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))
	tmp = 0
	if t <= -6.067626527897145e-58:
		tmp = t_1 + (t / (z * (y * 3.0)))
	elif t <= 3.830453512798634e-203:
		tmp = (x + ((y / 3.0) * (-1.0 / z))) + ((t / y) / (z * 3.0))
	else:
		tmp = t_1 + (0.3333333333333333 / (y * (z / t)))
	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(x - Float64(y / Float64(z * 3.0)))
	tmp = 0.0
	if (t <= -6.067626527897145e-58)
		tmp = Float64(t_1 + Float64(t / Float64(z * Float64(y * 3.0))));
	elseif (t <= 3.830453512798634e-203)
		tmp = Float64(Float64(x + Float64(Float64(y / 3.0) * Float64(-1.0 / z))) + Float64(Float64(t / y) / Float64(z * 3.0)));
	else
		tmp = Float64(t_1 + Float64(0.3333333333333333 / Float64(y * Float64(z / t))));
	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));
	tmp = 0.0;
	if (t <= -6.067626527897145e-58)
		tmp = t_1 + (t / (z * (y * 3.0)));
	elseif (t <= 3.830453512798634e-203)
		tmp = (x + ((y / 3.0) * (-1.0 / z))) + ((t / y) / (z * 3.0));
	else
		tmp = t_1 + (0.3333333333333333 / (y * (z / t)));
	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[(x - N[(y / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.067626527897145e-58], N[(t$95$1 + N[(t / N[(z * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.830453512798634e-203], N[(N[(x + N[(N[(y / 3.0), $MachinePrecision] * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t / y), $MachinePrecision] / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(0.3333333333333333 / N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	3.6
Target	1.7
Herbie	1.0

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

Derivation

1. Split input into 3 regimes

2. if t < -6.0676265278971449e-58

   1. Initial program 0.9

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

   2. Applied egg-rr 0.8

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{0 + z \cdot \left(3 \cdot y\right)}} \]

   if -6.0676265278971449e-58 < t < 3.8304535127986342e-203

   1. Initial program 6.8

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

   2. Applied egg-rr 1.1

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{1}{y} \cdot \left(t \cdot \frac{0.3333333333333333}{z}\right)} \]

   3. Applied egg-rr 0.2

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

   4. Applied egg-rr 0.2

      \[\leadsto \left(x - \color{blue}{\frac{1}{z} \cdot \frac{y}{3}}\right) + \frac{\frac{t}{y}}{z \cdot 3} \]

   if 3.8304535127986342e-203 < t

   1. Initial program 2.5

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

   2. Taylor expanded in z around 0 2.4

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

   3. Simplified 2.5

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\color{blue}{y \cdot \frac{z}{0.3333333333333333}}} \]
      Proof
      (*.f64 y (/.f64 z 1/3)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y z) 1/3)): 35 points increase in error, 37 points decrease in error
      (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (*.f64 y z))) 1/3): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 (Rewrite<= metadata-eval (neg.f64 -1)) (*.f64 y z)) 1/3): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/l*_binary64 (/.f64 (neg.f64 -1) (/.f64 1/3 (*.f64 y z)))): 23 points increase in error, 25 points decrease in error
      (Rewrite=> associate-/r/_binary64 (*.f64 (/.f64 (neg.f64 -1) 1/3) (*.f64 y z))): 22 points increase in error, 61 points decrease in error
      (*.f64 (/.f64 (Rewrite=> metadata-eval 1) 1/3) (*.f64 y z)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> metadata-eval 3) (*.f64 y z)): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in t around 0 2.5

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{0.3333333333333333 \cdot \frac{t}{y \cdot z}} \]

   5. Simplified 1.8

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{\frac{0.3333333333333333}{y \cdot \frac{z}{t}}} \]
      Proof
      (/.f64 1/3 (*.f64 y (/.f64 z t))): 0 points increase in error, 0 points decrease in error
      (/.f64 1/3 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y z) t))): 57 points increase in error, 36 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 1/3 t) (*.f64 y z))): 31 points increase in error, 30 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 1/3 (/.f64 t (*.f64 y z)))): 29 points increase in error, 33 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.067626527897145 \cdot 10^{-58}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{z \cdot \left(y \cdot 3\right)}\\ \mathbf{elif}\;t \leq 3.830453512798634 \cdot 10^{-203}:\\ \;\;\;\;\left(x + \frac{y}{3} \cdot \frac{-1}{z}\right) + \frac{\frac{t}{y}}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{0.3333333333333333}{y \cdot \frac{z}{t}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.9
Cost	1352

\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+17}:\\ \;\;\;\;t_1 + \frac{t}{3 \cdot \left(y \cdot z\right)}\\ \mathbf{elif}\;t \leq 3.830453512798634 \cdot 10^{-203}:\\ \;\;\;\;\left(x + \frac{y}{3} \cdot \frac{-1}{z}\right) + \frac{\frac{t}{y}}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{0.3333333333333333}{y \cdot \frac{z}{t}}\\ \end{array} \]

Alternative 2
Error	31.5
Cost	1244

\[\begin{array}{l} t_1 := \frac{y}{z \cdot -3}\\ \mathbf{if}\;y \leq -1.222067839580321 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-231}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-248}:\\ \;\;\;\;\frac{t}{y} \cdot \frac{0.3333333333333333}{z}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-224}:\\ \;\;\;\;\frac{\frac{t \cdot 0.3333333333333333}{z}}{y}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-140}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-133}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{elif}\;y \leq 2.302119268894705 \cdot 10^{+118}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	17.3
Cost	1240

\[\begin{array}{l} t_1 := \frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ t_2 := x + \frac{y}{z} \cdot -0.3333333333333333\\ \mathbf{if}\;y \leq -3.774279631201573 \cdot 10^{-59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{-167}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-231}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-224}:\\ \;\;\;\;\frac{\frac{t \cdot 0.3333333333333333}{z}}{y}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-140}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	1.6
Cost	1224

\[\begin{array}{l} t_1 := \frac{\frac{t}{y}}{z \cdot 3} + \left(x - \frac{\frac{y}{z}}{3}\right)\\ \mathbf{if}\;y \leq -1.9473759778457414 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 10^{-130}:\\ \;\;\;\;x + \frac{\frac{t \cdot 0.3333333333333333}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	1.2
Cost	1224

\[\begin{array}{l} t_1 := \left(x - \frac{y}{z \cdot 3}\right) + \frac{0.3333333333333333}{y \cdot \frac{z}{t}}\\ \mathbf{if}\;t \leq -5.8073270674823 \cdot 10^{-57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.830453512798634 \cdot 10^{-203}:\\ \;\;\;\;\frac{\frac{t}{y}}{z \cdot 3} + \left(x - \frac{\frac{y}{z}}{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	1.0
Cost	1224

\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+30}:\\ \;\;\;\;t_1 + \frac{t}{3 \cdot \left(y \cdot z\right)}\\ \mathbf{elif}\;t \leq 3.830453512798634 \cdot 10^{-203}:\\ \;\;\;\;\frac{\frac{t}{y}}{z \cdot 3} + \left(x - \frac{\frac{y}{z}}{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{0.3333333333333333}{y \cdot \frac{z}{t}}\\ \end{array} \]

Alternative 7
Error	1.0
Cost	1224

\[\begin{array}{l} t_1 := x - \frac{y}{z \cdot 3}\\ \mathbf{if}\;t \leq -6.067626527897145 \cdot 10^{-58}:\\ \;\;\;\;t_1 + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{elif}\;t \leq 3.830453512798634 \cdot 10^{-203}:\\ \;\;\;\;\frac{\frac{t}{y}}{z \cdot 3} + \left(x - \frac{\frac{y}{z}}{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{0.3333333333333333}{y \cdot \frac{z}{t}}\\ \end{array} \]

Alternative 8
Error	6.5
Cost	1104

\[\begin{array}{l} t_1 := x + \frac{\frac{t \cdot 0.3333333333333333}{z}}{y}\\ t_2 := x + \frac{y}{z} \cdot -0.3333333333333333\\ \mathbf{if}\;y \leq -3.717358786075202 \cdot 10^{-51}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 9.642077405213534 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.71199073711611 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{t}{y} - y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 1.0181362460642468 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	6.5
Cost	1104

\[\begin{array}{l} t_1 := x + \frac{y}{z} \cdot -0.3333333333333333\\ \mathbf{if}\;y \leq -3.717358786075202 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.642077405213534 \cdot 10^{-35}:\\ \;\;\;\;x + \frac{\frac{t \cdot 0.3333333333333333}{z}}{y}\\ \mathbf{elif}\;y \leq 9.71199073711611 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{t}{y} - y}{z \cdot 3}\\ \mathbf{elif}\;y \leq 1.0181362460642468 \cdot 10^{+36}:\\ \;\;\;\;x + \frac{0.3333333333333333}{\frac{z}{\frac{t}{y}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	1.6
Cost	968

\[\begin{array}{l} t_1 := x + \frac{-0.3333333333333333}{z} \cdot \left(y - \frac{t}{y}\right)\\ \mathbf{if}\;y \leq -1.9473759778457414 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-137}:\\ \;\;\;\;x + \frac{\frac{t \cdot 0.3333333333333333}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	29.0
Cost	848

\[\begin{array}{l} t_1 := \frac{y}{z \cdot -3}\\ \mathbf{if}\;x \leq -1.2270237265183216 \cdot 10^{+97}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.251499112551486 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.735234440178702 \cdot 10^{-278}:\\ \;\;\;\;\frac{t}{z} \cdot \frac{0.3333333333333333}{y}\\ \mathbf{elif}\;x \leq 6.2744705569812585 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	29.0
Cost	848

\[\begin{array}{l} t_1 := \frac{y}{z \cdot -3}\\ \mathbf{if}\;x \leq -1.2270237265183216 \cdot 10^{+97}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.251499112551486 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.735234440178702 \cdot 10^{-278}:\\ \;\;\;\;\frac{\frac{t \cdot 0.3333333333333333}{z}}{y}\\ \mathbf{elif}\;x \leq 6.2744705569812585 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	29.0
Cost	848

\[\begin{array}{l} t_1 := \frac{y}{z \cdot -3}\\ \mathbf{if}\;x \leq -1.2270237265183216 \cdot 10^{+97}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.251499112551486 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.735234440178702 \cdot 10^{-278}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t}{y \cdot z}\\ \mathbf{elif}\;x \leq 6.2744705569812585 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	11.4
Cost	840

\[\begin{array}{l} t_1 := x + \frac{y}{z} \cdot -0.3333333333333333\\ \mathbf{if}\;x \leq -4.2872561187635605 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.261237052647515 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{t}{y} - y}{z \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	28.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.2270237265183216 \cdot 10^{+97}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.2744705569812585 \cdot 10^{+29}:\\ \;\;\;\;\frac{y}{z \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16
Error	38.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022316 
(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))))