Main:z from

Average Error: 5.2 → 5.2

Time: 30.3s

Precision: binary64

Cost: 52672

Math

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

↓

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

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

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 (((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));
}

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 (((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))

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(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

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 = (((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

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[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]

Target

Original	5.2
Target	1.3
Herbie	5.2

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

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. Final simplification 5.2

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

Alternatives

Alternative 1
Error	27.5
Cost	39880

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ t_2 := \sqrt{1 + t} - \sqrt{t}\\ t_3 := \left(t_2 + \left(\left(\sqrt{1 + y} - \sqrt{y}\right) + \left(t_1 - \sqrt{z}\right)\right)\right) - -1\\ \mathbf{if}\;x \leq 3.2 \cdot 10^{-25}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{+20}:\\ \;\;\;\;t_2 + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\left(1 + t_1\right) - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	28.6
Cost	39748

\[\begin{array}{l} \mathbf{if}\;z \leq 2.6 \cdot 10^{+19}:\\ \;\;\;\;\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(\left(\sqrt{1 + x} - \sqrt{x}\right) + \left(\left(1 + \sqrt{1 + z}\right) - \sqrt{z}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y - -1} - \sqrt{y}\right)\\ \end{array} \]

Alternative 3
Error	20.8
Cost	39748

\[\begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-36}:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(\sqrt{1 + z} - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y - -1} - \sqrt{y}\right)\\ \end{array} \]

Alternative 4
Error	35.5
Cost	26696

\[\begin{array}{l} \mathbf{if}\;z \leq 2.6 \cdot 10^{-28}:\\ \;\;\;\;3 + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+19}:\\ \;\;\;\;\left(\sqrt{1 + z} + \sqrt{1 + x}\right) + \left(1 - \left(\sqrt{z} + \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y - -1} - \sqrt{y}\right)\\ \end{array} \]

Alternative 5
Error	35.5
Cost	26696

\[\begin{array}{l} \mathbf{if}\;z \leq 5.5 \cdot 10^{-31}:\\ \;\;\;\;3 + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+19}:\\ \;\;\;\;\left(\sqrt{1 + z} + \sqrt{1 + y}\right) + \left(1 - \left(\sqrt{z} + \sqrt{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y - -1} - \sqrt{y}\right)\\ \end{array} \]

Alternative 6
Error	35.2
Cost	13776

\[\begin{array}{l} t_1 := 1 + \left(\sqrt{t - -1} - \sqrt{t}\right)\\ \mathbf{if}\;z \leq 0.4:\\ \;\;\;\;3\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+189}:\\ \;\;\;\;2\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+238}:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	34.8
Cost	13512

\[\begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-28}:\\ \;\;\;\;3 + \left(\sqrt{1 + t} - \sqrt{t}\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+19}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y - -1} - \sqrt{y}\right)\\ \end{array} \]

Alternative 8
Error	32.9
Cost	13380

\[\begin{array}{l} \mathbf{if}\;z \leq 0.4:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y - -1} - \sqrt{y}\right)\\ \end{array} \]

Alternative 9
Error	32.5
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{-36}:\\ \;\;\;\;2 + \left(\sqrt{1 + z} - \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{y - -1} - \sqrt{y}\right)\\ \end{array} \]

Alternative 10
Error	46.9
Cost	196

\[\begin{array}{l} \mathbf{if}\;t \leq 0.0285:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;3\\ \end{array} \]

Alternative 11
Error	36.0
Cost	196

\[\begin{array}{l} \mathbf{if}\;z \leq 0.44:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 12
Error	41.3
Cost	64

\[3 \]

Reproduce

herbie shell --seed 2023075 
(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))))