Main:bigenough3 from C

Average Error: 29.8 → 29.2

Time: 8.8s

Precision: binary64

Cost: 26944

Math

\[\sqrt{x + 1} - \sqrt{x} \]

↓

\[\left(1 - \frac{\sqrt{x + 1} - \sqrt{x}}{-2}\right) + \left(\sqrt{1 + x} \cdot 0.5 - \left(\sqrt{x} \cdot 0.5 - -1\right)\right) \]

FPCore

(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))

↓

(FPCore (x)
 :precision binary64
 (+
  (- 1.0 (/ (- (sqrt (+ x 1.0)) (sqrt x)) -2.0))
  (- (* (sqrt (+ 1.0 x)) 0.5) (- (* (sqrt x) 0.5) -1.0))))

C

double code(double x) {
	return sqrt((x + 1.0)) - sqrt(x);
}

↓

double code(double x) {
	return (1.0 - ((sqrt((x + 1.0)) - sqrt(x)) / -2.0)) + ((sqrt((1.0 + x)) * 0.5) - ((sqrt(x) * 0.5) - -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x + 1.0d0)) - sqrt(x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - ((sqrt((x + 1.0d0)) - sqrt(x)) / (-2.0d0))) + ((sqrt((1.0d0 + x)) * 0.5d0) - ((sqrt(x) * 0.5d0) - (-1.0d0)))
end function

Java

public static double code(double x) {
	return Math.sqrt((x + 1.0)) - Math.sqrt(x);
}

↓

public static double code(double x) {
	return (1.0 - ((Math.sqrt((x + 1.0)) - Math.sqrt(x)) / -2.0)) + ((Math.sqrt((1.0 + x)) * 0.5) - ((Math.sqrt(x) * 0.5) - -1.0));
}

Python

def code(x):
	return math.sqrt((x + 1.0)) - math.sqrt(x)

↓

def code(x):
	return (1.0 - ((math.sqrt((x + 1.0)) - math.sqrt(x)) / -2.0)) + ((math.sqrt((1.0 + x)) * 0.5) - ((math.sqrt(x) * 0.5) - -1.0))

Julia

function code(x)
	return Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
end

↓

function code(x)
	return Float64(Float64(1.0 - Float64(Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) / -2.0)) + Float64(Float64(sqrt(Float64(1.0 + x)) * 0.5) - Float64(Float64(sqrt(x) * 0.5) - -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((x + 1.0)) - sqrt(x);
end

↓

function tmp = code(x)
	tmp = (1.0 - ((sqrt((x + 1.0)) - sqrt(x)) / -2.0)) + ((sqrt((1.0 + x)) * 0.5) - ((sqrt(x) * 0.5) - -1.0));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[(1.0 - N[(N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] - N[(N[(N[Sqrt[x], $MachinePrecision] * 0.5), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	29.8
Target	0.2
Herbie	29.2

\[\frac{1}{\sqrt{x + 1} + \sqrt{x}} \]

Derivation

1. Initial program 29.8

   \[\sqrt{x + 1} - \sqrt{x} \]

2. Applied egg-rr 29.9

   \[\leadsto \color{blue}{\left(1 - \frac{\sqrt{x + 1} - \sqrt{x}}{-2}\right) + \left(\left(\sqrt{x + 1} - \sqrt{x}\right) \cdot 0.5 + -1\right)} \]

3. Applied egg-rr 29.2

   \[\leadsto \left(1 - \frac{\sqrt{x + 1} - \sqrt{x}}{-2}\right) + \color{blue}{\left(\sqrt{1 + x} \cdot 0.5 - \left(\sqrt{x} \cdot 0.5 - -1\right)\right)} \]

4. Final simplification 29.2

   \[\leadsto \left(1 - \frac{\sqrt{x + 1} - \sqrt{x}}{-2}\right) + \left(\sqrt{1 + x} \cdot 0.5 - \left(\sqrt{x} \cdot 0.5 - -1\right)\right) \]

Alternatives

Alternative 1
Error	29.8
Cost	26688

\[\begin{array}{l} t_0 := \sqrt{1 + x}\\ 0.5 \cdot \left(\left(\left(\left(1 + t_0\right) + -1\right) - \left(\sqrt{x} - t_0\right)\right) - \sqrt{x}\right) \end{array} \]

Alternative 2
Error	29.8
Cost	13120

\[\sqrt{x + 1} - \sqrt{x} \]

Alternative 3
Error	30.8
Cost	6848

\[0.5 \cdot x + \left(1 - \sqrt{x}\right) \]

Alternative 4
Error	55.8
Cost	64

\[0.5 \]

Alternative 5
Error	31.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023074 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))