Main:z from

Average Error: 5.2 → 5.1

Time: 28.8s

Precision: binary64

Cost: 118592

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

Math

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

↓

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

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 z)) (sqrt z)) (sqrt x))
          (- (sqrt (+ 1.0 t)) (sqrt t)))))
   (+
    (+ (sqrt (+ x 1.0)) (- (sqrt (+ 1.0 y)) (sqrt y)))
    (* t_1 (* t_1 (/ 1.0 t_1))))))

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 + z)) - sqrt(z)) - sqrt(x)) + (sqrt((1.0 + t)) - sqrt(t));
	return (sqrt((x + 1.0)) + (sqrt((1.0 + y)) - sqrt(y))) + (t_1 * (t_1 * (1.0 / t_1)));
}

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
    real(8) :: t_1
    t_1 = ((sqrt((1.0d0 + z)) - sqrt(z)) - sqrt(x)) + (sqrt((1.0d0 + t)) - sqrt(t))
    code = (sqrt((x + 1.0d0)) + (sqrt((1.0d0 + y)) - sqrt(y))) + (t_1 * (t_1 * (1.0d0 / t_1)))
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) {
	double t_1 = ((Math.sqrt((1.0 + z)) - Math.sqrt(z)) - Math.sqrt(x)) + (Math.sqrt((1.0 + t)) - Math.sqrt(t));
	return (Math.sqrt((x + 1.0)) + (Math.sqrt((1.0 + y)) - Math.sqrt(y))) + (t_1 * (t_1 * (1.0 / t_1)));
}

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 + z)) - math.sqrt(z)) - math.sqrt(x)) + (math.sqrt((1.0 + t)) - math.sqrt(t))
	return (math.sqrt((x + 1.0)) + (math.sqrt((1.0 + y)) - math.sqrt(y))) + (t_1 * (t_1 * (1.0 / t_1)))

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(Float64(Float64(sqrt(Float64(1.0 + z)) - sqrt(z)) - sqrt(x)) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t)))
	return Float64(Float64(sqrt(Float64(x + 1.0)) + Float64(sqrt(Float64(1.0 + y)) - sqrt(y))) + Float64(t_1 * Float64(t_1 * Float64(1.0 / t_1))))
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 + z)) - sqrt(z)) - sqrt(x)) + (sqrt((1.0 + t)) - sqrt(t));
	tmp = (sqrt((x + 1.0)) + (sqrt((1.0 + y)) - sqrt(y))) + (t_1 * (t_1 * (1.0 / t_1)));
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[(N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(t$95$1 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	5.2
Target	0.4
Herbie	5.1

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

Derivation

1. Initial program 5.2

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

2. Simplified 18.9

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

   [Start] 5.2	\[ \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 5.2	\[ \color{blue}{\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-82 [=>] 5.2	\[ \color{blue}{\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 5.2	\[ \color{blue}{\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 5.2	\[ \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\left(\left(\sqrt{y + 1} - \sqrt{y}\right) + \left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-108 [=>] 5.2	\[ \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \color{blue}{\left(\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) - \sqrt{x}\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-108 [=>] 5.2	\[ \color{blue}{\left(\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right)\right) - \sqrt{x}} \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 5.2	\[ \color{blue}{\left(\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right)} - \sqrt{x} \]
   rational_best_oopsla_all_46_json_45_simplify-107 [=>] 5.2	\[ \color{blue}{\left(\sqrt{x + 1} + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\left(\left(\sqrt{t + 1} - \sqrt{t}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) - \sqrt{x}\right)} \]

3. Applied egg-rr 5.1

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

4. Final simplification 5.1

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

Alternatives

Alternative 1
Error	5.2
Cost	52672

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

Alternative 2
Error	5.6
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{x + 1}\\ \mathbf{if}\;y \leq 4.5 \cdot 10^{-25}:\\ \;\;\;\;\left(t_1 - \sqrt{x}\right) + \left(\left(\sqrt{1 + z} - \sqrt{z}\right) + \left(1 + \left(\sqrt{1 + t} - \sqrt{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(-\sqrt{x}\right)\\ \end{array} \]

Alternative 3
Error	5.2
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 3.8 \cdot 10^{+29}:\\ \;\;\;\;\left(\left(1 + t_1\right) - \sqrt{y}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(t_1 - \sqrt{y}\right)\right) + \left(-\sqrt{x}\right)\\ \end{array} \]

Alternative 4
Error	5.6
Cost	26696

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 10^{-20}:\\ \;\;\;\;\left(\sqrt{1 + t} - \left(\sqrt{z} + \sqrt{t}\right)\right) + \left(2 + t_1\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+14}:\\ \;\;\;\;1 + \left(t_1 + \left(t_2 - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(t_2 - \sqrt{y}\right)\right) + \left(-\sqrt{x}\right)\\ \end{array} \]

Alternative 5
Error	9.4
Cost	26568

\[\begin{array}{l} t_1 := \sqrt{1 + x}\\ \mathbf{if}\;y \leq 3 \cdot 10^{-25}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+22}:\\ \;\;\;\;t_1 + \left(\sqrt{1 + y} - \left(\sqrt{x} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - \sqrt{x}\\ \end{array} \]

Alternative 6
Error	8.9
Cost	26564

\[\begin{array}{l} t_1 := \sqrt{1 + y}\\ \mathbf{if}\;z \leq 2.6 \cdot 10^{+14}:\\ \;\;\;\;1 + \left(\sqrt{1 + z} + \left(t_1 - \left(\sqrt{y} + \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(t_1 - \sqrt{y}\right)\right) + \left(-\sqrt{x}\right)\\ \end{array} \]

Alternative 7
Error	9.4
Cost	26500

\[\begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-25}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x + 1} + \left(\sqrt{1 + y} - \sqrt{y}\right)\right) + \left(-\sqrt{x}\right)\\ \end{array} \]

Alternative 8
Error	9.5
Cost	20040

\[\begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-25}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+22}:\\ \;\;\;\;\left(\sqrt{1 + y} + 1\right) - \left(\sqrt{x} + \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 9
Error	9.6
Cost	13512

\[\begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-25}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+22}:\\ \;\;\;\;1 + \left(\sqrt{y - -1} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 10
Error	22.7
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+24}:\\ \;\;\;\;1 + \left(\sqrt{y - -1} - \sqrt{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 11
Error	23.8
Cost	13252

\[\begin{array}{l} \mathbf{if}\;y \leq 2.9:\\ \;\;\;\;\left(2 + 0.5 \cdot y\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 12
Error	24.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 7.2:\\ \;\;\;\;\left(2 + 0.5 \cdot y\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{x}\\ \end{array} \]

Alternative 13
Error	24.2
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq 5.8:\\ \;\;\;\;\left(2 + 0.5 \cdot y\right) - \sqrt{y}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x + 1\right) - \sqrt{x}\\ \end{array} \]

Alternative 14
Error	25.8
Cost	6724

\[\begin{array}{l} \mathbf{if}\;y \leq 9.6:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{x}\\ \end{array} \]

Alternative 15
Error	25.8
Cost	6724

\[\begin{array}{l} \mathbf{if}\;y \leq 15:\\ \;\;\;\;2 - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{x}\\ \end{array} \]

Alternative 16
Error	25.8
Cost	196

\[\begin{array}{l} \mathbf{if}\;y \leq 4.5:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Error	41.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023090 
(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))))