Average Error: 5.0 → 2.3
Time: 9.6s
Precision: binary64
\[ \begin{array}{c}[x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \end{array} \]
\[\left(\left(\left(\sqrt{x + 1} - \sqrt{x}\right) + \left(\sqrt{y + 1} - \sqrt{y}\right)\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]
↓
\[\left(\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \log \left(e^{\sqrt{1 + z} - \sqrt{z}}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]
(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
 (+
  (+
   (+
    (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x)))
    (log1p (expm1 (- (sqrt (+ 1.0 y)) (sqrt y)))))
   (log (exp (- (sqrt (+ 1.0 z)) (sqrt z)))))
  (- (sqrt (+ 1.0 t)) (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));
}
↓
double code(double x, double y, double z, double t) {
	return (((1.0 / (sqrt((1.0 + x)) + sqrt(x))) + log1p(expm1((sqrt((1.0 + y)) - sqrt(y))))) + log(exp((sqrt((1.0 + z)) - sqrt(z))))) + (sqrt((1.0 + t)) - 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));
}
↓
public static double code(double x, double y, double z, double t) {
	return (((1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x))) + Math.log1p(Math.expm1((Math.sqrt((1.0 + y)) - Math.sqrt(y))))) + Math.log(Math.exp((Math.sqrt((1.0 + z)) - Math.sqrt(z))))) + (Math.sqrt((1.0 + t)) - 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))
↓
def code(x, y, z, t):
	return (((1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))) + math.log1p(math.expm1((math.sqrt((1.0 + y)) - math.sqrt(y))))) + math.log(math.exp((math.sqrt((1.0 + z)) - math.sqrt(z))))) + (math.sqrt((1.0 + t)) - math.sqrt(t))

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(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x))) + log1p(expm1(Float64(sqrt(Float64(1.0 + y)) - sqrt(y))))) + log(exp(Float64(sqrt(Float64(1.0 + z)) - sqrt(z))))) + Float64(sqrt(Float64(1.0 + t)) - sqrt(t)))
end

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[(1.0 / N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Log[1 + N[(Exp[N[(N[Sqrt[N[(1.0 + y), $MachinePrecision]], $MachinePrecision] - N[Sqrt[y], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Log[N[Exp[N[(N[Sqrt[N[(1.0 + z), $MachinePrecision]], $MachinePrecision] - N[Sqrt[z], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(1.0 + t), $MachinePrecision]], $MachinePrecision] - N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\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(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \log \left(e^{\sqrt{1 + z} - \sqrt{z}}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right)

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original5.0
Target0.4
Herbie2.3
\[\left(\left(\frac{1}{\sqrt{x + 1} + \sqrt{x}} + \frac{1}{\sqrt{y + 1} + \sqrt{y}}\right) + \frac{1}{\sqrt{z + 1} + \sqrt{z}}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

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-rr4.7

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

  3. Taylor expanded in x around 0 2.3

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

  4. Applied egg-rr2.3

    \[\leadsto \left(\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{y + 1} - \sqrt{y}\right)\right)}\right) + \left(\sqrt{z + 1} - \sqrt{z}\right)\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

  5. Applied egg-rr2.3

    \[\leadsto \left(\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{y + 1} - \sqrt{y}\right)\right)\right) + \color{blue}{\log \left(e^{\sqrt{1 + z} - \sqrt{z}}\right)}\right) + \left(\sqrt{t + 1} - \sqrt{t}\right) \]

  6. Final simplification2.3

    \[\leadsto \left(\left(\frac{1}{\sqrt{1 + x} + \sqrt{x}} + \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{1 + y} - \sqrt{y}\right)\right)\right) + \log \left(e^{\sqrt{1 + z} - \sqrt{z}}\right)\right) + \left(\sqrt{1 + t} - \sqrt{t}\right) \]

Reproduce

herbie shell --seed 2022170 
(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))))