Main:z from

Average Error: 5.0 → 1.3

Time: 10.4s

Precision: binary64

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

Math

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

↓

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

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 y)) (sqrt y)))
        (t_2 (+ (sqrt (+ 1.0 x)) (sqrt x))))
   (+
    (+ (* (/ 1.0 t_2) (/ (+ t_2 t_1) t_1)) (- (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) {
	double t_1 = sqrt((1.0 + y)) + sqrt(y);
	double t_2 = sqrt((1.0 + x)) + sqrt(x);
	return (((1.0 / t_2) * ((t_2 + t_1) / t_1)) + (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
    real(8) :: t_1
    real(8) :: t_2
    t_1 = sqrt((1.0d0 + y)) + sqrt(y)
    t_2 = sqrt((1.0d0 + x)) + sqrt(x)
    code = (((1.0d0 / t_2) * ((t_2 + t_1) / t_1)) + (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) {
	double t_1 = Math.sqrt((1.0 + y)) + Math.sqrt(y);
	double t_2 = Math.sqrt((1.0 + x)) + Math.sqrt(x);
	return (((1.0 / t_2) * ((t_2 + t_1) / t_1)) + (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):
	t_1 = math.sqrt((1.0 + y)) + math.sqrt(y)
	t_2 = math.sqrt((1.0 + x)) + math.sqrt(x)
	return (((1.0 / t_2) * ((t_2 + t_1) / t_1)) + (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)
	t_1 = Float64(sqrt(Float64(1.0 + y)) + sqrt(y))
	t_2 = Float64(sqrt(Float64(1.0 + x)) + sqrt(x))
	return Float64(Float64(Float64(Float64(1.0 / t_2) * Float64(Float64(t_2 + t_1) / t_1)) + 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)
	t_1 = sqrt((1.0 + y)) + sqrt(y);
	t_2 = sqrt((1.0 + x)) + sqrt(x);
	tmp = (((1.0 / t_2) * ((t_2 + t_1) / t_1)) + (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_] := 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[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(N[(t$95$2 + t$95$1), $MachinePrecision] / t$95$1), $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]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	5.0
Target	0.3
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 5.0

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

2. Applied egg-rr 4.7

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

3. Taylor expanded in x around 0 2.4

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

4. Applied egg-rr 1.3

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

5. Applied egg-rr 1.3

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

6. Final simplification 1.3

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

Reproduce

herbie shell --seed 2022160 
(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))))