Average Error: 0.3 → 0.3
Time: 10.9s
Precision: binary64
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
↓
\[\begin{array}{l} t_1 := \sqrt[3]{e^{t}}\\ \left(\left(0.5 \cdot x - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot {\left(\left|t_1\right|\right)}^{t}\right)\right) \cdot {\left(\sqrt{t_1}\right)}^{t} \end{array} \]
\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}}
↓
\begin{array}{l}
t_1 := \sqrt[3]{e^{t}}\\
\left(\left(0.5 \cdot x - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot {\left(\left|t_1\right|\right)}^{t}\right)\right) \cdot {\left(\sqrt{t_1}\right)}^{t}
\end{array}

(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
 (let* ((t_1 (cbrt (exp t))))
   (*
    (* (- (* 0.5 x) y) (* (sqrt (* 2.0 z)) (pow (fabs t_1) t)))
    (pow (sqrt t_1) 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) {
	double t_1 = cbrt(exp(t));
	return (((0.5 * x) - y) * (sqrt(2.0 * z) * pow(fabs(t_1), t))) * pow(sqrt(t_1), 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. Applied add-cube-cbrt_binary640.3

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

  4. Applied sqrt-prod_binary640.3

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

  5. Applied unpow-prod-down_binary640.3

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

  6. Applied associate-*r*_binary640.3

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

  7. Simplified0.3

    \[\leadsto \color{blue}{\left(\left(\left(0.5 \cdot x - y\right) \cdot \sqrt{2 \cdot z}\right) \cdot {\left(\left|\sqrt[3]{e^{t}}\right|\right)}^{t}\right)} \cdot {\left(\sqrt{\sqrt[3]{e^{t}}}\right)}^{t} \]

  8. Applied associate-*l*_binary640.3

    \[\leadsto \color{blue}{\left(\left(0.5 \cdot x - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot {\left(\left|\sqrt[3]{e^{t}}\right|\right)}^{t}\right)\right)} \cdot {\left(\sqrt{\sqrt[3]{e^{t}}}\right)}^{t} \]

  9. Final simplification0.3

    \[\leadsto \left(\left(0.5 \cdot x - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot {\left(\left|\sqrt[3]{e^{t}}\right|\right)}^{t}\right)\right) \cdot {\left(\sqrt{\sqrt[3]{e^{t}}}\right)}^{t} \]

Reproduce

herbie shell --seed 2022088 
(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))))