Average Error: 0.3 → 0.3
Time: 8.3s
Precision: binary64
\[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
↓
\[\begin{array}{l} t_1 := \sqrt[3]{\sqrt{e^{t}}}\\ \left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(t_1 \cdot t_1\right)}^{t}\right) \cdot \left({\left(\sqrt[3]{e^{t \cdot 0.3333333333333333}}\right)}^{t} \cdot {\left(\sqrt[3]{\sqrt{\sqrt[3]{e^{t}}}}\right)}^{t}\right) \end{array} \]
\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]{\sqrt{e^{t}}}\\
\left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(t_1 \cdot t_1\right)}^{t}\right) \cdot \left({\left(\sqrt[3]{e^{t \cdot 0.3333333333333333}}\right)}^{t} \cdot {\left(\sqrt[3]{\sqrt{\sqrt[3]{e^{t}}}}\right)}^{t}\right)
\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 (sqrt (exp t)))))
   (*
    (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (* t_1 t_1) t))
    (*
     (pow (cbrt (exp (* t 0.3333333333333333))) t)
     (pow (cbrt (sqrt (cbrt (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) {
	double t_1 = cbrt(sqrt(exp(t)));
	return ((((x * 0.5) - y) * sqrt(z * 2.0)) * pow((t_1 * t_1), t)) * (pow(cbrt(exp(t * 0.3333333333333333)), t) * pow(cbrt(sqrt(cbrt(exp(t)))), 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

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

  4. Applied unpow-prod-down_binary640.3

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

  5. Applied associate-*r*_binary640.3

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

  6. Applied add-cube-cbrt_binary640.3

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

  7. Applied cbrt-prod_binary640.3

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

  8. Applied unpow-prod-down_binary640.3

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

  9. Simplified0.3

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

  10. Simplified0.3

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

  11. Taylor expanded in t around inf 0.3

    \[\leadsto \left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(\sqrt[3]{\sqrt{e^{t}}} \cdot \sqrt[3]{\sqrt{e^{t}}}\right)}^{t}\right) \cdot \left({\left(\sqrt[3]{\color{blue}{e^{0.3333333333333333 \cdot t}}}\right)}^{t} \cdot {\left(\sqrt[3]{\sqrt{\sqrt[3]{e^{t}}}}\right)}^{t}\right) \]

  12. Final simplification0.3

    \[\leadsto \left(\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(\sqrt[3]{\sqrt{e^{t}}} \cdot \sqrt[3]{\sqrt{e^{t}}}\right)}^{t}\right) \cdot \left({\left(\sqrt[3]{e^{t \cdot 0.3333333333333333}}\right)}^{t} \cdot {\left(\sqrt[3]{\sqrt{\sqrt[3]{e^{t}}}}\right)}^{t}\right) \]

Reproduce

herbie shell --seed 2021275 
(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))))