Main:z from

Average Error: 5.4 → 5.4

Time: 35.6s

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	5.4
Target	1.5
Herbie	5.4

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

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. Final simplification 5.4

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

Alternatives

Alternative 1
Error	24.4
Cost	39748

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 1.66:\\ \;\;\;\;\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\left(\sqrt{1 + t} - \sqrt{t}\right) + \left(1 + \left(t_1 - \sqrt{z}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - -1} - \left(\sqrt{x} - \left(\left(1 + t_1\right) - \sqrt{z}\right)\right)\\ \end{array} \]

Alternative 2
Error	41.4
Cost	26820

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 0.09:\\ \;\;\;\;1 + \left(\left(t_1 - \left(\left(\sqrt{z} - \sqrt{1 + x}\right) - -1\right)\right) + \left(1 - \sqrt{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - -1} - \left(\sqrt{x} - \left(\left(1 + t_1\right) - \sqrt{z}\right)\right)\\ \end{array} \]

Alternative 3
Error	29.7
Cost	26564

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

Alternative 4
Error	36.8
Cost	26564

\[\begin{array}{l} t_1 := \sqrt{1 + z}\\ \mathbf{if}\;y \leq 1.4:\\ \;\;\;\;\left(t_1 - \sqrt{z}\right) - -2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x - -1} - \left(\sqrt{x} - \left(\left(1 + t_1\right) - \sqrt{z}\right)\right)\\ \end{array} \]

Alternative 5
Error	36.5
Cost	14040

\[\begin{array}{l} t_1 := \sqrt{z - -1}\\ t_2 := \left(t_1 + 2\right) - \sqrt{z}\\ \mathbf{if}\;y \leq 8.5 \cdot 10^{-253}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-206}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-138}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-110}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 0.072:\\ \;\;\;\;\left(1 + \sqrt{1 + x}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(t_1 - \sqrt{z}\right)\\ \end{array} \]

Alternative 6
Error	33.1
Cost	13776

\[\begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-171}:\\ \;\;\;\;3\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-110}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-82}:\\ \;\;\;\;1 + \left(1 + \left(\left(1 + z \cdot 0.5\right) - \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 0.35:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{z - -1} - \sqrt{z}\right)\\ \end{array} \]

Alternative 7
Error	34.9
Cost	13776

\[\begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-171}:\\ \;\;\;\;3\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-110}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{-82}:\\ \;\;\;\;1 + \left(1 + \left(\left(1 + z \cdot 0.5\right) - \sqrt{z}\right)\right)\\ \mathbf{elif}\;y \leq 0.24:\\ \;\;\;\;\left(1 + \sqrt{1 + x}\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{z - -1} - \sqrt{z}\right)\\ \end{array} \]

Alternative 8
Error	29.8
Cost	13380

\[\begin{array}{l} \mathbf{if}\;y \leq 0.58:\\ \;\;\;\;\left(\sqrt{1 + z} - \sqrt{z}\right) - -2\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\sqrt{z - -1} - \sqrt{z}\right)\\ \end{array} \]

Alternative 9
Error	35.7
Cost	7236

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

Alternative 10
Error	35.8
Cost	6980

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

Alternative 11
Error	36.0
Cost	196

\[\begin{array}{l} \mathbf{if}\;z \leq 0.235:\\ \;\;\;\;3\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 12
Error	36.5
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2023077 
(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))))