Average Error: 0.3 → 0.3
Time: 13.9s
Precision: binary64
Cost: 19968
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
\[\left(x \cdot 0.5 - y\right) \cdot \sqrt{{\left(e^{t}\right)}^{t} \cdot \left(2 \cdot z\right)} \]
(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
 (* (- (* x 0.5) y) (sqrt (* (pow (exp t) t) (* 2.0 z)))))
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 ((x * 0.5) - y) * sqrt((pow(exp(t), t) * (2.0 * z)));
}
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 = ((x * 0.5d0) - y) * sqrt(((exp(t) ** t) * (2.0d0 * z)))
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 ((x * 0.5) - y) * Math.sqrt((Math.pow(Math.exp(t), t) * (2.0 * z)));
}
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 ((x * 0.5) - y) * math.sqrt((math.pow(math.exp(t), t) * (2.0 * z)))
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(x * 0.5) - y) * sqrt(Float64((exp(t) ^ t) * Float64(2.0 * z))))
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 = ((x * 0.5) - y) * sqrt(((exp(t) ^ t) * (2.0 * z)));
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[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(N[Power[N[Exp[t], $MachinePrecision], t], $MachinePrecision] * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}
\left(x \cdot 0.5 - y\right) \cdot \sqrt{{\left(e^{t}\right)}^{t} \cdot \left(2 \cdot z\right)}

Error

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

    \[\leadsto \color{blue}{\left(x \cdot 0.5 - y\right) \cdot \left(\sqrt{z \cdot 2} \cdot \sqrt{e^{t \cdot t}}\right)} \]
    Proof
    (*.f64 (-.f64 (*.f64 x 1/2) y) (*.f64 (sqrt.f64 (*.f64 z 2)) (sqrt.f64 (exp.f64 (*.f64 t t))))): 0 points increase in error, 0 points decrease in error
    (*.f64 (-.f64 (*.f64 x 1/2) y) (*.f64 (sqrt.f64 (*.f64 z 2)) (Rewrite<= exp-sqrt_binary64 (exp.f64 (/.f64 (*.f64 t t) 2))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 (*.f64 x 1/2) y) (sqrt.f64 (*.f64 z 2))) (exp.f64 (/.f64 (*.f64 t t) 2)))): 0 points increase in error, 3 points decrease in error
  3. Applied egg-rr34.8

    \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\left(z \cdot 2\right) \cdot e^{t \cdot t}}\right)} - 1\right)} \]
  4. Simplified0.3

    \[\leadsto \left(x \cdot 0.5 - y\right) \cdot \color{blue}{\sqrt{\left(2 \cdot z\right) \cdot {\left(e^{t}\right)}^{t}}} \]
    Proof
    (*.f64 (-.f64 (*.f64 x 1/2) y) (sqrt.f64 (*.f64 (*.f64 2 z) (pow.f64 (exp.f64 t) t)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (-.f64 (*.f64 x 1/2) y) (sqrt.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 z 2)) (pow.f64 (exp.f64 t) t)))): 1 points increase in error, 1 points decrease in error
    (*.f64 (-.f64 (*.f64 x 1/2) y) (sqrt.f64 (*.f64 (*.f64 z 2) (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 t t)))))): 1 points increase in error, 4 points decrease in error
    (*.f64 (-.f64 (*.f64 x 1/2) y) (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (sqrt.f64 (*.f64 (*.f64 z 2) (exp.f64 (*.f64 t t)))))))): 0 points increase in error, 1 points decrease in error
    (*.f64 (-.f64 (*.f64 x 1/2) y) (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (sqrt.f64 (*.f64 (*.f64 z 2) (exp.f64 (*.f64 t t)))))) 1))): 1 points increase in error, 1 points decrease in error
  5. Final simplification0.3

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

Alternatives

Alternative 1
Error0.3
Cost13632
\[\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot e^{t \cdot t}} \]
Alternative 2
Error0.8
Cost7488
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot \left(1 + 0.5 \cdot \left(t \cdot t\right)\right) \]
Alternative 3
Error0.8
Cost7360
\[\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot \left(z + z \cdot \left(t \cdot t\right)\right)} \]
Alternative 4
Error0.8
Cost7360
\[\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)} \]
Alternative 5
Error1.2
Cost6976
\[\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z} \]
Alternative 6
Error32.2
Cost6720
\[x \cdot \sqrt{0.5 \cdot z} \]

Error

Reproduce

herbie shell --seed 2022343 
(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))))