Main:z from

Average Error: 5.4 → 0.1

Time: 36.1s

Precision: binary64

Cost: 92864

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

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{1 + y} + \sqrt{y}\\ t_2 := \sqrt{x} + \mathsf{hypot}\left(1, \sqrt{x}\right)\\ \left(\frac{t_1 + t_2 \cdot \left(1 + \left(y - y\right)\right)}{t_1 \cdot t_2} + \frac{1 + \left(z - z\right)}{\sqrt{1 + z} + \sqrt{z}}\right) + \frac{1 + \left(t - t\right)}{\sqrt{1 + t} + \sqrt{t}} \end{array} \]

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
 (let* ((t_1 (+ (sqrt (+ 1.0 y)) (sqrt y)))
        (t_2 (+ (sqrt x) (hypot 1.0 (sqrt x)))))
   (+
    (+
     (/ (+ t_1 (* t_2 (+ 1.0 (- y y)))) (* t_1 t_2))
     (/ (+ 1.0 (- z z)) (+ (sqrt (+ 1.0 z)) (sqrt z))))
    (/ (+ 1.0 (- t t)) (+ (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) {
	double t_1 = sqrt((1.0 + y)) + sqrt(y);
	double t_2 = sqrt(x) + hypot(1.0, sqrt(x));
	return (((t_1 + (t_2 * (1.0 + (y - y)))) / (t_1 * t_2)) + ((1.0 + (z - z)) / (sqrt((1.0 + z)) + sqrt(z)))) + ((1.0 + (t - t)) / (sqrt((1.0 + t)) + sqrt(t)));
}

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) {
	double t_1 = Math.sqrt((1.0 + y)) + Math.sqrt(y);
	double t_2 = Math.sqrt(x) + Math.hypot(1.0, Math.sqrt(x));
	return (((t_1 + (t_2 * (1.0 + (y - y)))) / (t_1 * t_2)) + ((1.0 + (z - z)) / (Math.sqrt((1.0 + z)) + Math.sqrt(z)))) + ((1.0 + (t - t)) / (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):
	t_1 = math.sqrt((1.0 + y)) + math.sqrt(y)
	t_2 = math.sqrt(x) + math.hypot(1.0, math.sqrt(x))
	return (((t_1 + (t_2 * (1.0 + (y - y)))) / (t_1 * t_2)) + ((1.0 + (z - z)) / (math.sqrt((1.0 + z)) + math.sqrt(z)))) + ((1.0 + (t - t)) / (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)
	t_1 = Float64(sqrt(Float64(1.0 + y)) + sqrt(y))
	t_2 = Float64(sqrt(x) + hypot(1.0, sqrt(x)))
	return Float64(Float64(Float64(Float64(t_1 + Float64(t_2 * Float64(1.0 + Float64(y - y)))) / Float64(t_1 * t_2)) + Float64(Float64(1.0 + Float64(z - z)) / Float64(sqrt(Float64(1.0 + z)) + sqrt(z)))) + Float64(Float64(1.0 + Float64(t - t)) / 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)
	t_1 = sqrt((1.0 + y)) + sqrt(y);
	t_2 = sqrt(x) + hypot(1.0, sqrt(x));
	tmp = (((t_1 + (t_2 * (1.0 + (y - y)))) / (t_1 * t_2)) + ((1.0 + (z - z)) / (sqrt((1.0 + z)) + sqrt(z)))) + ((1.0 + (t - t)) / (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_] := Block[{t$95$1 = N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] + N[Sqrt[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(t$95$1 + N[(t$95$2 * N[(1.0 + N[(y - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(z - z), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] + N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(t - t), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] + N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	5.4
Target	0.4
Herbie	0.1

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

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 2.6

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

3. Applied egg-rr 1.3

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

4. Applied egg-rr 0.4

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

5. Applied egg-rr 0.1

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

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.2
Cost	92292

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + y}\\ t_3 := t_2 - \sqrt{y}\\ t_4 := t_2 + \sqrt{y}\\ t_5 := \sqrt{1 + t}\\ t_6 := \sqrt{1 + x}\\ \mathbf{if}\;t_3 \leq 0.002:\\ \;\;\;\;\left(t_5 - \sqrt{t}\right) + \left(\left(t_1 - \sqrt{z}\right) - \frac{\sqrt{x} + \left(t_4 + t_6\right)}{t_4 \cdot \left(\left(-\sqrt{x}\right) - t_6\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(t - t\right)}{t_5 + \sqrt{t}} + \left(\frac{1 + \left(z - z\right)}{t_1 + \sqrt{z}} + \left(t_3 + \left(t_6 - \sqrt{x}\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	0.4
Cost	79552

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

Alternative 3
Error	0.4
Cost	79488

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

Alternative 4
Error	0.2
Cost	79300

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + z}\\ t_3 := \sqrt{1 + y}\\ t_4 := t_3 + \sqrt{y}\\ t_5 := \sqrt{1 + t}\\ \mathbf{if}\;y \leq 0.9702627827596261:\\ \;\;\;\;\frac{1 + \left(t - t\right)}{t_5 + \sqrt{t}} + \left(\frac{1 + \left(z - z\right)}{t_2 + \sqrt{z}} + \left(\left(t_3 - \sqrt{y}\right) + \left(t_1 - \sqrt{x}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_5 - \sqrt{t}\right) + \left(\frac{\sqrt{x} + \left(t_4 + t_1\right)}{t_4} \cdot \frac{1}{\sqrt{x} + t_1} + \left(t_2 - \sqrt{z}\right)\right)\\ \end{array} \]

Alternative 5
Error	1.2
Cost	66372

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + x}\\ t_3 := \sqrt{1 + t} - \sqrt{t}\\ t_4 := \sqrt{1 + y}\\ t_5 := t_4 + \sqrt{y}\\ \mathbf{if}\;t_2 - \sqrt{x} \leq 0.9999999999995:\\ \;\;\;\;t_3 + \left(\left(t_1 - \sqrt{z}\right) + \left(\left(t_4 - \sqrt{y}\right) + \frac{1 + \left(x - x\right)}{\sqrt{x} + t_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 + \left(\frac{1 + \left(z - z\right)}{t_1 + \sqrt{z}} + \frac{1 + t_5}{t_5}\right)\\ \end{array} \]

Alternative 6
Error	1.5
Cost	53824

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

Alternative 7
Error	1.9
Cost	53440

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

Alternative 8
Error	2.6
Cost	53056

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

Alternative 9
Error	2.8
Cost	40132

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;y \leq 109664145554519.2:\\ \;\;\;\;t_2 + \left(\left(t_1 - \sqrt{z}\right) + \left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(\frac{1 + \left(z - z\right)}{t_1 + \sqrt{z}} + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \end{array} \]

Alternative 10
Error	2.0
Cost	40132

\[\begin{array}{l} t_1 := \frac{1 + \left(z - z\right)}{\sqrt{1 + z} + \sqrt{z}}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;y \leq 109664145554519.2:\\ \;\;\;\;t_2 + \left(t_1 + \left(\sqrt{1 + y} + \left(1 - \sqrt{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(t_1 + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \end{array} \]

Alternative 11
Error	6.5
Cost	39752

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 2.4090741330030487 \cdot 10^{-29}:\\ \;\;\;\;1 + \left(1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(t_1 - \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 274302974449287040:\\ \;\;\;\;\left(\left(\sqrt{1 + z} - \sqrt{y}\right) + \left(t_1 + \left(t_2 - \sqrt{x}\right)\right)\right) - \sqrt{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_2 - \sqrt{y}\right)\\ \end{array} \]

Alternative 12
Error	6.5
Cost	39752

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 2.4090741330030487 \cdot 10^{-29}:\\ \;\;\;\;1 + \left(1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(t_1 - \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 274302974449287040:\\ \;\;\;\;\left(\sqrt{1 + z} - \left(\sqrt{y} + \sqrt{z}\right)\right) + \left(t_1 + \left(t_2 - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_2 - \sqrt{y}\right)\\ \end{array} \]

Alternative 13
Error	2.8
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{1 + z} - \sqrt{z}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ \mathbf{if}\;y \leq 5.7273362715610585 \cdot 10^{+20}:\\ \;\;\;\;t_2 + \left(t_1 + \left(\left(1 + \sqrt{1 + y}\right) - \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(t_1 + \frac{1}{\sqrt{x} + \sqrt{1 + x}}\right)\\ \end{array} \]

Alternative 14
Error	6.5
Cost	26696

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 2.4090741330030487 \cdot 10^{-29}:\\ \;\;\;\;1 + \left(1 + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + x} - \sqrt{x}\right)\right)\right)\\ \mathbf{elif}\;z \leq 274302974449287040:\\ \;\;\;\;\left(1 + \left(t_1 + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 - \sqrt{y}\right)\\ \end{array} \]

Alternative 15
Error	9.9
Cost	26564

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 274302974449287040:\\ \;\;\;\;\left(1 + \left(t_1 + \sqrt{1 + z}\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 - \sqrt{y}\right)\\ \end{array} \]

Alternative 16
Error	11.2
Cost	20164

\[\begin{array}{l} \mathbf{if}\;z \leq 0.9407274237819818:\\ \;\;\;\;\left(2 + \left(\sqrt{1 + x} + z \cdot 0.5\right)\right) - \left(\sqrt{x} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\\ \end{array} \]

Alternative 17
Error	11.2
Cost	20164

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 0.9407274237819818:\\ \;\;\;\;\left(2 + \left(t_1 + z \cdot 0.5\right)\right) - \left(\sqrt{y} + \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 - \sqrt{y}\right)\\ \end{array} \]

Alternative 18
Error	22.9
Cost	13248

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

Alternative 19
Error	41.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022318 
(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))))