Average Error: 0.3 → 0.3
Time: 10.8s
Precision: binary64
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
↓
\[\left(\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot {\left(\sqrt{e^{t}}\right)}^{t} \]
(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))
↓
(FPCore (x y z t)
 :precision binary64
 (* (* (- (* 0.5 x) y) (sqrt (* 2.0 z))) (pow (sqrt (exp t)) t)))
double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
}
↓
double code(double x, double y, double z, double t) {
	return (((0.5 * x) - y) * sqrt((2.0 * z))) * pow(sqrt(exp(t)), t);
}

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 = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * exp(((t * t) / 2.0d0))
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 = (((0.5d0 * x) - y) * sqrt((2.0d0 * z))) * (sqrt(exp(t)) ** t)
end function

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.exp(((t * t) / 2.0));
}
↓
public static double code(double x, double y, double z, double t) {
	return (((0.5 * x) - y) * Math.sqrt((2.0 * z))) * Math.pow(Math.sqrt(Math.exp(t)), t);
}

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.exp(((t * t) / 2.0))
↓
def code(x, y, z, t):
	return (((0.5 * x) - y) * math.sqrt((2.0 * z))) * math.pow(math.sqrt(math.exp(t)), t)

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * exp(Float64(Float64(t * t) / 2.0)))
end
↓
function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(0.5 * x) - y) * sqrt(Float64(2.0 * z))) * (sqrt(exp(t)) ^ t))
end

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
end
↓
function tmp = code(x, y, z, t)
	tmp = (((0.5 * x) - y) * sqrt((2.0 * z))) * (sqrt(exp(t)) ^ t);
end

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_] := N[(N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sqrt[N[Exp[t], $MachinePrecision]], $MachinePrecision], t], $MachinePrecision]), $MachinePrecision]

\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}
↓
\left(\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot {\left(\sqrt{e^{t}}\right)}^{t}

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

Original0.3
Target0.3
Herbie0.3
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{1}\right)}^{\left(\frac{t \cdot t}{2}\right)} \]

Derivation

  1. Initial program 0.3

    \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]

  2. Simplified0.3

    \[\leadsto \color{blue}{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(\sqrt{e^{t}}\right)}^{t}} \]

  3. Taylor expanded in x around 0 0.6

    \[\leadsto \color{blue}{\left(0.5 \cdot \left(\left(\sqrt{2} \cdot x\right) \cdot \sqrt{z}\right) - \left(y \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)} \cdot {\left(\sqrt{e^{t}}\right)}^{t} \]

  4. Simplified0.5

    \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \left(\sqrt{z} \cdot \left(0.5 \cdot x - y\right)\right)\right)} \cdot {\left(\sqrt{e^{t}}\right)}^{t} \]

  5. Applied egg-rr0.3

    \[\leadsto \color{blue}{{\left(\left(\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot {\left(\sqrt{e^{t}}\right)}^{t}\right)}^{1}} \]

  6. Final simplification0.3

    \[\leadsto \left(\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot {\left(\sqrt{e^{t}}\right)}^{t} \]

Reproduce

herbie shell --seed 2022133 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :herbie-target
  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp 1.0) (/ (* t t) 2.0)))

  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))