Main:z from

Average Error: 5.4 → 1.3

Time: 33.7s

Precision: binary64

Cost: 53440

\[ \begin{array}{c}[x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \end{array} \]

Math

\[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

↓

\[\left(\left(\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}} + \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y)))
   (- (sqrt (+ z 1.0)) (sqrt z)))
  (- (sqrt (+ t 1.0)) (sqrt t))))

↓

(FPCore (x y z t)
 :precision binary64
 (+
  (+
   (+
    (/ (+ 1.0 (- x x)) (+ (sqrt (+ 1.0 x)) (sqrt x)))
    (/ (+ 1.0 (- y y)) (+ (sqrt (+ 1.0 y)) (sqrt y))))
   (- (sqrt (+ 1.0 z)) (sqrt z)))
  (- (sqrt (+ 1.0 t)) (sqrt t))))

C

double code(double x, double y, double z, double t) {
	return (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
}

↓

double code(double x, double y, double z, double t) {
	return ((((1.0 + (x - x)) / (sqrt((1.0 + x)) + sqrt(x))) + ((1.0 + (y - y)) / (sqrt((1.0 + y)) + sqrt(y)))) + (sqrt((1.0 + z)) - sqrt(z))) + (sqrt((1.0 + t)) - sqrt(t));
}

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 = (((sqrt((x + 1.0d0)) - sqrt(x)) + (sqrt((y + 1.0d0)) - sqrt(y))) + (sqrt((z + 1.0d0)) - sqrt(z))) + (sqrt((t + 1.0d0)) - sqrt(t))
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
    code = ((((1.0d0 + (x - x)) / (sqrt((1.0d0 + x)) + sqrt(x))) + ((1.0d0 + (y - y)) / (sqrt((1.0d0 + y)) + sqrt(y)))) + (sqrt((1.0d0 + z)) - sqrt(z))) + (sqrt((1.0d0 + t)) - sqrt(t))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((Math.sqrt((x + 1.0)) - Math.sqrt(x)) + (Math.sqrt((y + 1.0)) - Math.sqrt(y))) + (Math.sqrt((z + 1.0)) - Math.sqrt(z))) + (Math.sqrt((t + 1.0)) - Math.sqrt(t));
}

↓

public static double code(double x, double y, double z, double t) {
	return ((((1.0 + (x - x)) / (Math.sqrt((1.0 + x)) + Math.sqrt(x))) + ((1.0 + (y - y)) / (Math.sqrt((1.0 + y)) + Math.sqrt(y)))) + (Math.sqrt((1.0 + z)) - Math.sqrt(z))) + (Math.sqrt((1.0 + t)) - Math.sqrt(t));
}

Python

def code(x, y, z, t):
	return (((math.sqrt((x + 1.0)) - math.sqrt(x)) + (math.sqrt((y + 1.0)) - math.sqrt(y))) + (math.sqrt((z + 1.0)) - math.sqrt(z))) + (math.sqrt((t + 1.0)) - math.sqrt(t))

↓

def code(x, y, z, t):
	return ((((1.0 + (x - x)) / (math.sqrt((1.0 + x)) + math.sqrt(x))) + ((1.0 + (y - y)) / (math.sqrt((1.0 + y)) + math.sqrt(y)))) + (math.sqrt((1.0 + z)) - math.sqrt(z))) + (math.sqrt((1.0 + t)) - math.sqrt(t))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) + Float64(sqrt(Float64(y + 1.0)) - sqrt(y))) + Float64(sqrt(Float64(z + 1.0)) - sqrt(z))) + Float64(sqrt(Float64(t + 1.0)) - sqrt(t)))
end

↓

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) + Float64(Float64(1.0 + Float64(y - y)) / Float64(sqrt(Float64(1.0 + y)) + sqrt(y)))) + Float64(sqrt(Float64(1.0 + z)) - sqrt(z))) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((sqrt((x + 1.0)) - sqrt(x)) + (sqrt((y + 1.0)) - sqrt(y))) + (sqrt((z + 1.0)) - sqrt(z))) + (sqrt((t + 1.0)) - sqrt(t));
end

↓

function tmp = code(x, y, z, t)
	tmp = ((((1.0 + (x - x)) / (sqrt((1.0 + x)) + sqrt(x))) + ((1.0 + (y - y)) / (sqrt((1.0 + y)) + sqrt(y)))) + (sqrt((1.0 + z)) - sqrt(z))) + (sqrt((1.0 + t)) - sqrt(t));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(y + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(z + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(t + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(y - y), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	5.4
Target	0.4
Herbie	1.3

\[\left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

Derivation

1. Initial program 5.4

   \[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

2. Applied egg-rr 2.5

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

3. Applied egg-rr 1.3

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

4. Final simplification 1.3

   \[\leadsto \left(\left(\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}} + \frac{1 + \left(y - y\right)}{\sqrt{1 + y} + \sqrt{y}}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

Alternatives

Alternative 1
Error	2.4
Cost	66116

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ t_2 := \sqrt{1 + x}\\ \mathbf{if}\;\left(t_1 - \sqrt{y}\right) + \left(t_2 - \sqrt{x}\right) \leq 1:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{t_2 + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \frac{1}{t_1 + \sqrt{y}}\right)\right)\\ \end{array} \]

Alternative 2
Error	2.5
Cost	53056

\[\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) \]

Alternative 3
Error	2.8
Cost	39748

\[\begin{array}{l} \mathbf{if}\;y \leq 6.881410638406484 \cdot 10^{+25}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]

Alternative 4
Error	6.6
Cost	39492

\[\begin{array}{l} t_1 := {\left(1 + x\right)}^{0.25}\\ \mathbf{if}\;z \leq 11242347565169738:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(2 + \sqrt{1 + z}\right) - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_1, t_1, \sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \end{array} \]

Alternative 5
Error	7.1
Cost	39300

\[\begin{array}{l} \mathbf{if}\;z \leq 442930321498721.25:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(2 + \sqrt{1 + z}\right) - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\sqrt{1 + x} + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)}\right)}^{3}\\ \end{array} \]

Alternative 6
Error	14.4
Cost	26964

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \frac{1 + \left(x - x\right)}{t_1 + \sqrt{x}}\\ t_3 := \left(t_1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{if}\;z \leq 1.5170240156713598 \cdot 10^{-40}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + 2\right)\\ \mathbf{elif}\;z \leq 11242347565169738:\\ \;\;\;\;\left(2 + \sqrt{1 + z}\right) - \sqrt{z}\\ \mathbf{elif}\;z \leq 7.339617824165878 \cdot 10^{+113}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.5520218046905906 \cdot 10^{+142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 6.764936828573377 \cdot 10^{+149}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	14.1
Cost	26832

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \frac{1 + \left(x - x\right)}{t_1 + \sqrt{x}}\\ t_3 := \left(t_1 + \sqrt{1 + y}\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{if}\;z \leq 11242347565169738:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(2 + \sqrt{1 + z}\right) - \sqrt{z}\right)\\ \mathbf{elif}\;z \leq 7.339617824165878 \cdot 10^{+113}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.5520218046905906 \cdot 10^{+142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 6.764936828573377 \cdot 10^{+149}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	18.6
Cost	13908

\[\begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ t_2 := 1 + t_1\\ \mathbf{if}\;z \leq 1.5170240156713598 \cdot 10^{-40}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + 2\right)\\ \mathbf{elif}\;z \leq 442930321498721.25:\\ \;\;\;\;\left(2 + \sqrt{1 + z}\right) - \sqrt{z}\\ \mathbf{elif}\;z \leq 7.339617824165878 \cdot 10^{+113}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.564835403114244 \cdot 10^{+129}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 6.764936828573377 \cdot 10^{+149}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	16.3
Cost	13900

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;z \leq 1.5170240156713598 \cdot 10^{-40}:\\ \;\;\;\;1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + 2\right)\\ \mathbf{elif}\;z \leq 442930321498721.25:\\ \;\;\;\;\left(2 + \sqrt{1 + z}\right) - \sqrt{z}\\ \mathbf{elif}\;z \leq 7.339617824165878 \cdot 10^{+113}:\\ \;\;\;\;1 + \left(t_1 - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{t_1 + \sqrt{x}}\\ \end{array} \]

Alternative 10
Error	21.7
Cost	13776

\[\begin{array}{l} t_1 := \sqrt{1 + x} - \sqrt{x}\\ t_2 := 1 + t_1\\ \mathbf{if}\;z \leq 442930321498721.25:\\ \;\;\;\;\left(2 + \sqrt{1 + z}\right) - \sqrt{z}\\ \mathbf{elif}\;z \leq 7.339617824165878 \cdot 10^{+113}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.564835403114244 \cdot 10^{+129}:\\ \;\;\;\;1\\ \mathbf{elif}\;z \leq 6.764936828573377 \cdot 10^{+149}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	26.3
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 1.6161217900490776 \cdot 10^{+32}:\\ \;\;\;\;\sqrt{1 + z} + \left(2 - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 12
Error	26.3
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 1.6161217900490776 \cdot 10^{+32}:\\ \;\;\;\;\left(2 + \sqrt{1 + z}\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 13
Error	41.4
Cost	13120

\[\sqrt{1 + x} - \sqrt{x} \]

Alternative 14
Error	42.2
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022316 
(FPCore (x y z t)
  :name "Main:z from "
  :precision binary64

  :herbie-target
  (+ (+ (+ (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))) (/ 1.0 (+ (sqrt (+ y 1.0)) (sqrt y)))) (/ 1.0 (+ (sqrt (+ z 1.0)) (sqrt z)))) (- (sqrt (+ t 1.0)) (sqrt t)))

  (+ (+ (+ (- (sqrt (+ x 1.0)) (sqrt x)) (- (sqrt (+ y 1.0)) (sqrt y))) (- (sqrt (+ z 1.0)) (sqrt z))) (- (sqrt (+ t 1.0)) (sqrt t))))