Average Error: 0.3 → 0.3

Time: 11.5s

Precision: binary64

Cost: 26368

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

↓

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

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

↓

{\left(\sqrt{e^{t}}\right)}^{t} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{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
 (* (pow (sqrt (exp t)) t) (* (- (* x 0.5) y) (sqrt (* 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 pow(sqrt(exp(t)), t) * (((x * 0.5) - y) * sqrt(2.0 * z));
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	0.3
Target	0.3
Herbie	0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	13760

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

Alternative 2
Error	0.9
Cost	7488

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

Alternative 3
Error	1.3
Cost	6976

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

Alternative 4
Error	61.3
Cost	64

\[0\]

Alternative 5
Error	61.7
Cost	64

\[1\]

Error

Derivation

Initial program 0.3
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}\]
Simplified0.3
\[\leadsto \color{blue}{\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(\sqrt{e^{t}}\right)}^{t}}\]
Simplified0.3
\[\leadsto \color{blue}{{\left(\sqrt{e^{t}}\right)}^{t} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right)}\]
Final simplification0.3
\[\leadsto {\left(\sqrt{e^{t}}\right)}^{t} \cdot \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{2 \cdot z}\right)\]

Reproduce

herbie shell --seed 2021044 
(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))))